Role of critical stress in quantifying the magnitude of fluid-injection triggered earthquakes

Domain: earth_science
Published: 2024-09-10
Editorial outcome: minor_revision

Abstract
Here we define and report the relationship between the maximum seismic magnitude (M) and injection volume (ΔV) through fluid-injection fault-reactivation experiments and analysis. This relationship incorporates the in situ shear modulus (G) and fault pre-stress as a fraction of the strength drop (c), expressed as M = c/(1-c)GΔV. Injection response defines a sigmoidal relation in M-ΔV space with unit gradient limbs linked by an intermediate up-step. Both laboratory observations and analysis for a rigid fault with slip limited to the zone of pressurization show trajectories of cumulative M-ΔV that evolve at a gradient of unity, are offset in order of increasing pre-stress and are capable of step changes in moment with shear reactivation at elevated critical-stresses – key features apparent in field observations. The model and confirmatory laboratory observations explain the occurrence of some triggered earthquakes at EGS sites significantly larger than expected relative to injection volumes and based on previous models.

Introduction
While earthquakes that occur naturally are typically the result of the gradual accumulation of tectonic stresses over geological time, they can also be triggered by relatively small stress perturbations if the affected region is already close to failure. Triggered seismicity refers to earthquakes triggered by human activities and typically falls into the latter category1–3. The past fifteen years have seen a rapid increase in triggered earthquakes. These events result from massive reinjection of waste fluids from oil and gas extraction4–10, or for the stimulation of reservoirs as hydraulic fracturing11 or hydraulic shearing12,13. The spatial proximity of these seismicity to such industrial operations and the fact that many of these events are preceded by months to years of high-volume fluid injection suggest a connection between the occurrence of seismicity and fluid injection14–16. Fluid injection triggers seismicity through two distinct mechanisms: (i) elevated pore pressures reduce the strength of a tectonically primed fault with the adjacent stress state remaining essentially unchanged, or (ii) far-field stresses are perturbed beyond the immediate region of pore pressure diffusion with fault strength remaining largely un-affected17–19.

Importantly, triggered earthquakes can be of substantial magnitude and highly destructive, such as the M-5.5 2017 Pohang earthquake in South Korea that resulted in significant infrastructure damage and civilian injuries12,13,20. As such, these occurrences pose a potential threat in many branches of energy production and utilization – particularly those for net-zero-carbon (e.g., EGS, hot sedimentary reservoirs, CO2 sequestration)21–24 and reduced-carbon (e.g., gas shale, coalbed methane) alternatives to conventional fossil fuels (i.e., coal and conventional oil)25. Such fluid-triggered earthquakes are observed to be causally linked to both overpressures and pressurization rates7,14,15,18,26,27, to evolve from both creeping slip28 and poroelastic stress projections3,11 and capable of characterization as both contained11,29–31 and runaway ruptures2,32–34. Laboratory experiments confirm various controls of pressure and pressurization rates35–37, pressure distributions38, evolution of permeability29,39, presence of asperities40 and the role of pre-stress29,31,37,38,41 in driving slip and influencing the magnitude of the largest anticipated earthquake - a key but elusive parameter in evaluating seismic hazard. To first order, seismic moment has been shown to scale with injection volume for both constrained17,29–31 and runaway ruptures2,32. These relationships link maximum seismic moment (M) to the product of the total injected fluid volume (ΔV) and the shear modulus (G) of the affected zone as M = αGΔVβ where α and β represent the pre-stress (α) and the form of the rupture as either constrained to within the over-pressurized patch or runaway beyond it (β). Despite their differences, they broadly align with current observations of the M-ΔV relationships within the range of 2 < M < 631 but remain equivocal in both defining the gradient of M = αGΔVβ as a power of either β=1 or 3/2, and in representing large events triggered by meager injection volumes such as the 2017 M-5.5 Pohang earthquake.

We define and quantify the role of pre-existing shear stress in controlling the form (gradient) and magnitude of the M-ΔV relationship through a suite of experiments on laboratory faults constrained by end-member conditions of zero-displacement and constant stress and triggered by fluid overpressures. We define and observe a sigmoidal response in M-ΔV space comprising pre-slip, slip then post-slip phases whose magnitudes and gradients honor field observations. We explain our laboratory observations with a rupture model that accommodates pre-stress on a fault constrained to fail within the zone of fluid pressurization to allow parameterization of the model and extension to behavior at field scale.

Results
We perform shear reactivation experiments on laboratory faults pre-loaded close to failure and reactivated by the injection of fluid into the fault. The sample comprises a single-inclined-fracture (SIF) transecting a cylindrical sample (2.5-cm diameter by 5-cm long, Fig. 1) of Westerly granite with fluid pressure supplied to the fault by small conduits (drill holes) piercing the up-dip and down-dip extent of the fault. The artificial fault plane is inclined at 30° to the longitudinal axis of the core and polished with #60 grit grinding powder subsequently reworked through repeated shearing tests. The reactivation experiments introduce small-scale roughness on the fault surface, simulating the natural roughness of a fault that has experienced repeated slip events.

All experiments are at ambient temperature and follow a similar protocol. This involves (i) application of confining stresses (3 MPa) on the fault fully saturated with de-ionized water, followed by (ii) shear-mobilization through the increase of axial loading at a constant displacement rate until a post-peak steady-state condition is reached. (iii) The axial loading and related shear stress are reduced to a prescribed fraction of the peak steady-state frictional strength (typically 60% to 90%, representing intermediate to high magnitudes). (iv) Fault reactivation is triggered by stepwise increasing the pore pressure on the fault in 0.1 MPa increments held constant for 1–5-minutes. Uniform pressure on the fault is confirmed by the equilibration of the upstream and downstream pressure transducers and typically occurs in 1 to 20 secs – depending on the fault permeability, modulated by the applied fluid pressure.

This study considers two boundary conditions related to the in-situ stress state: constant shear stress (CSS) and zero-displacement (ZD). CSS represents a simplified model of a fault under constant tectonic shear stress, achieved by keeping the confining stress (σ3) and axial stress (σ1) constant, which results in a constant shear stress (τ). In contrast, ZD represents the case where the elastic strain energy in the (testing) system dissipates as the fault slips, causing the fault to relax and the axial/shear stress to decrease during displacement. The shearing of faults/fractures and associated stress relaxation are widely observed in episodes of seismicity or microseismicity34,42. To achieve ZD, the axial loading pump is locked, causing the differential stress (σd = σ1 - σ3) acting on the sample to decrease significantly in response to the sample shortening with the gradual sliding of the fault, and therefore a decrease in shear stress. As apparent in Eq. (2), relaxation of the axial stress also results in a decrease in the effective normal stress (σ'n), and thus a reduction in the fault frictional strength, which may promote slip. Slip will further reduce shear stress via system stiffness until the fault self-arrests. In other words, under ZD boundary conditions, the fault slip behavior is determined by the competition between the reduced fault strength that promotes slip and the reduced shear stress that limits slip.

Linking seismic moment to injected volume
The cumulative moment is recovered from the shear stress drop (Δτ) driven at each reactivation within the sample of total volume (V)43 as:
M = ΔτV = Δu kt V (3)
where kt is the tangential system stiffness (kt = Δτ/Δu) calibrated from the elastic stiffness of a locked fault [Pa/m] and Δu represents the tangential displacement (i.e., parallel to the fault). The loading tangential stiffness is measured during the initial phase of shear mobilization, when the axial stress is increased from a hydrostatic state to its peak strength (as illustrated in Fig. 2a–c).

Cumulative injection volume into the fault is estimated assuming a uniform dilation across the rigid fault wall as a function of the uniformly distributed fluid pressurization, and defined in terms of a constant normal stiffness

Methods
The sample assembly is jacketed in double latex sleeves and confined within a triaxial pressure vessel. The latex sleeves retain the structural arrangement of the core inside the pressure vessel, isolate the sample from the confining fluid and confine the fluid that fills the interior of the core. The pressure vessel provides independent control on axial stress (pump A), confining stress (pump B) and upstream pore pressure (pump C). Axial displacement rates are set through the volume rates of pump A and confirmed by a linear variable differential transducer (LVDT) connected to the axial loading piston. Shear displacement Δu is then calculated as the along fracture component of displacement as Δu = Δua / cosθ where Δua is the axial displacement of the sample. Fault compaction/dilation-related displacements are neglected in this as their magnitudes are small relative to shear offset.

Two 1.3-mm boreholes drilled from each end of the cylindrical sample pierce the close extent of the fault plane (along the sidewall) and allow fluid injection along the fault. Sealing the downstream exit port enables a uniform fluid pressure distribution to be applied, finely incremented (+/−0.04 MPa) and measured both by the pump and a pressure transducer located downstream. Given the SIF configuration, the effective normal stress and shear stress can be recovered from the confining stress, axial stress and pore pressure as:
σ'n = σ3 - Pp + (σ1 - σ3) sin2θ (1)
τ = (σ1 - σ3) sinθcosθ (2)
where σ1 is axial stress, σ3 is confining pressure stress, Pp is pore pressure, and θ (30°) is the angle with respect to the longitudinal axis of the sample.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_2": "Review report for “Role of Critical Stress in Quantifying the Magnitude of Fluid-Injection Triggered Earthquakes” by Yu et al.\nOver the past decade or so, fluid-injection-related industrial operations have been performed for a variety of purposes, including wastewater disposal, hydraulic fracturing, enhanced geothermal. As a byproduct, these operations have also caused small-to-moderate earthquakes, sometimes in a relatively stable region and with an unexpected magnitude, thus raising serious concerns on regulation, safety issue, mental health, etc. As such, it is important to know the proper protocols or strategies during fluid injection operation, for example, how to prevent the occurrence of un-controlled, potentially runaway, earthquakes. To this end, Yu et al. have conducted a series of triaxial experiments, by injecting homogeneously-distributed fluid into a fully saturated and prestressed fault. They focus on the role of prestress in modulating the relation between the cumulative (or the maximum) seismic moment and the volume of injected fluid. They find that increasing the proximity of fault’s prestress to failure can reduce the amount of injected fluid required for fault reactivation. Equivalently, injecting the same amount of fluid into a fault with higher susceptibility to failure tends to produce a larger event. These findings hold true for two different types of loading conditions: constant shear stress (CSS) and zero displacement (ZD). Then, the authors propose a theoretical model for explaining their experimental observations and compare the results with other observations in the lab or in situ. Overall, this study provides a better estimation for the maximum moment of induced earthquakes than the conventional model of McGarr (JGR, 1976, 2014) with a fixed prestress level.\nOn one hand, I find this study interesting and suitable for publication in Nature Communications, as it presents both experimental results and theoretical analysis, which can have a direct and timely impact to the scientific and industrial communities. On the other hand, I am afraid the novelty of this study may be compromised, at least partly, by a recent study from the same research group (Li et al., Nature Communications, 2021, doi:10.1038/s41467-020-20700-4). If the authors can provide more evidence to show that this study includes significantly new understanding on the role of prestress, then I will be happy to make my further recommendation. Below I list my detailed concern on novelty as well as some other comments.\nMajor comments:\n(1) Comparison to previous studies\nSeveral previous studies have already shown that factors other than the total volume of injected fluid can influence the nature of fluid-injection-induced earthquakes, e.g., whether they are confined to the zone of pressurization or can expand beyond it. These factors include normal stress, initial shear stress, injection rate, and aseismic slip (e.g., Galis et al., Science Advances, 2017, doi:10.1126/sciadv.aap7528; Passelègue et al., GRL, 2018, doi:10.1029/2018GL080470; Bhattacharya and Viesca, Science, 2019, doi:10.1126/science.aaw7354; Ji and Wu, Tectonophysics, 2020, doi:10.1016/j.tecto.2020.228572). I think these studies should be appreciated in the introduction to tell readers the current stage of understanding fluid-injection-induced earthquakes. Especially, the work of Li et al. (Nature Communications, 2021, doi:10.1038/s41467-020-20700-4) should be discussed, not only because it was done by the same research group and published in the same journal, but also because it also investigated the role of prestress and proposed a theoretical model very similar to this study. As such, I feel the introduction part, e.g., lines 44-57, needs to be revised. The revised version should tell readers what have already been done (specifically on the role of prestress) and set up the stage for this study (e.g., what else can be done for improving the understanding of the role of prestress).\n(2) The uniqueness of this study\nFollowing (1) and regarding the relation to Li et al. (Nature Communications, 2021, doi:10.1038/s41467-020-20700-4), I think the authors should do a better job advertising the unique aspects (or findings) of their study. From my own reading, their experimental configuration (triaxial) is not exactly the same as that in Li et al. (double direct shear). Consequently, the results could be different. For example, during the CSS test of this study, the maintenance of constant shear stress implies a continued increase of differential stress (\\sigma_1 - \\sigma_3) to compensate the shear stress release due to slip. But this also means there is an increase of normal stress (the term after + in Eq. 1) to compensate the normal stress reduction due to fluid injection. This situation is different from that under a double direct shear in Li et al. (NC, 2021), where normal stress change is solely caused by fluid injection. Moreover, the temporal evolution of injection-induced displacement seems to be different: it shows decreasing slope under CSS and increasing slope under ZD in this study (Fig. 3), but shows increasing slope under CSS and decreasing slope under ZD in Li et al. (their figure 1). I am not sure if such contrasting results can be attributed to the difference in experimental configuration, but I do encourage the authors to think about the new findings of their study.\n(3) The estimation of shear displacement (lines 84-85)\nI wonder if the authors have made corrections for the elastic deformation of the rock sample and that of the loading machine. In principle, to estimate the true shear displacement along the fault, elastic deformation from various sources needs to be subtracted from the raw record of load-point displacement (e.g., see Ji and Wu, Tectonophysics, 2020, doi:10.1016/j.tecto.2020.228572).\nI also wonder if the authors have made corrections for the change of “actual” contact area during slip, as it is known that in typical triaxial test the “actual” contact area can decrease with slip.\n(4) Fig. 4 and lines 192-193\nI had a hard time in understanding the fault behavior during the so-called “first pre-slip stage”. According to the description in lines 192-193, fault is locked at this stage, but Fig. 4 shows that the cumulative seismic moment is not zero during this stage. So, either the authors forgot to make corrections for elastic deformation (see Comment #3), or the tangential stiffness is not a constant (as both shear stress and normal stress are increasing during axial loading). Some clarification is needed: whether there is only elastic deformation or small amount of fault slip during “pre-slip stage”.\n(5) Moment magnitude or moment\nIn many places (e.g., line 118, 174, 189, 190, 205, 283, 286, 290, “Event magnitude” in Fig. 3), the authors have incorrectly used “moment magnitude” for “moment (M0)”. Please correct the use.\n(6) Abnormal results for CSS4 (Table 1)\nCould the authors explain why CSS4 only caused 3760 micrometer cumulative displacement, despite a similar condition to CSS3? Plus, I also notice the stiffness value for CSS4 seems to be abnormal compared to other cases.\n(7) Assumption of nearly complete stress drop\nIn writing Eq. (9), nearly complete stress drop (from initial shear stress c*\\tau_s to zero residual shear stress) is assumed. Could the authors provide some rationale for supporting such assumption?\n(8) Line 241 and Eq. (13): In deriving this equation, a Gutenberg-Richter (G-R) distribution with b value at 1 has to be assumed, as in McGarr (JGR, 2014). But such distribution does not hold for the authors’ case, where the experimentally-generated events seem to follow a rather concentrated distribution (Fig. 3), perhaps related to the generation of characteristic earthquakes(?). I guess the underlying reasons are that events generated in the lab are controlled by machine stiffness and constant increment of fluid pressure (0.1 MPa, line 101), and that the experimental fault is rather uniform (not favorable for producing a G-R distribution with b around 1). I think these differences between laboratory settings and natural conditions should be emphasized in the text. These differences could complicate the comparison between laboratory events and natural events, so that some statements (e.g., lines 306-308) may have to be tuned down a bit.\nMinor comments:\nBoth “triggered” and “induced” (e.g., lines 1, 12, 16, and many other places) have been used in this study. I know different researchers may have different preferences, but to avoid confusion, it is better to clarify the meanings of “triggered” and “induced”.\nLine 25: please define G (shear modulus)\nLines 27-28: perhaps add “some” before “triggered earthquakes”? Also, the authors use “EGS sites” here. I guess they want to refer to the famous Pohang earthquake case, but are other cases (earthquakes with larger magnitudes than expected) also at EGS sites? If not, I would suggest removing “EGS sites”.\nLine 45: add “South Korea” to help understand the location of Pohang.\nLine 53: strictly speaking, \\alpha is not constant. In Ref. 28, it depends on a normalized prestress parameter c, similar to the current study. In Ref. 29, it also depends on the prestress level \\tau_0.\nLine 61: I am not sure if “zero displacement” is the correct terminology. In the context of earthquake science, we usually call it “constant displacement”. I leave it to the authors to decide if they want to make a change.\nLine 96: please be more explicit, DI = de-ionized\nLine 98: in “The axial loading and related shear stress is …”, is should be are\nLine 119-120: Eq. (3) may not be obvious to some readers, unless they are familiar with the concepts of transformational strain and stress glut, or the previous work of McGarr. Please add the source for Eq. (3).\nLines 155-157: the experimental configurations are different between Ref. 28 (double direct shear) and the current study (triaxial), I am not sure if the authors can simply refer to Ref. 28 to support the mentioned statement.\nFig. 3b: I can see a large-amplitude oscillation of fluid pressure around 3200 s, what has happened?\nLines 191-192: Fig. 4 is in log-log scale, meaning that any apparent linear relation could actually mean a power-law relation. If the authors want to highlight a linear relation between cumulative moment and fluid volume, it would be better to add some reference line in Fig. 4.\nFig. 5a focuses on the relation related to the maximum seismic moment. Please clarify if “Our data: CSS & ZD” refer to the actual data or the estimated relation based on Eq. (13).\nLine 271: the mentioned pre-stress to strength ratio of 60% to 90% does not match the one in In Fig. 5b&c (10% to 50%).\nLines 279-282: it is fine to suggest that shale may have a lower shear modulus; but in the authors’ case Westerly granite is used, whose shear modulus is not anomalously low.", "reviewer_3": "This study reports shear reactivation experiments on laboratory faults pre-loaded close to failure and reactivated by the injection of fluid into the fault. Similar experiments have been reported in many papers by the authors and other researchers, particularly Li et al, NC, (2021).\nOnly 2 zero-displacement experiments with a narrow range of critical stress (85% and 89%) were conducted. The relevant data points are far below the scaling lines in Fig. 5c, in comparison to the results in Fig. 5b, showing insignificant seismic risks related to the zero-displacement conditions. More zero-displacement experiments with a wide range of critical stress may need to show the significance of this condition.\nThe noteworthy outcome is the analytical scaling between cumulative moment magnitude and injected fluid volume with an additional consideration of pre-stress compared to Li et al (2021).\nThe cumulative seismic moment incorporating pre-stress is a smart idea and should be the main contribution of this study. However, the validation relying only on experimental data is not fully convincible. The lab-scale experiments cannot accommodate many field-scale seismic processes, such as fault ruptures. So, the validation with induced earthquakes (such as data points in Fig. 5a) is really needed to show the reliability of the scaling. Otherwise, this work is not of significance to the field."}, "author_response": "Dear Reviewers,\nWe appreciate the positive comments from the reviewers and the time that they have invested in improving our manuscript. Please see below our itemized responses with modifications applied separately to the marked-up manuscript.\nSpecifically, the revisions have:\n Emphasized the unique contribution of the paper.\n Clarified and explained several concepts and results.\nThe following identifies responses to the review comments. Review comments are shown in blue and responses in red. Changes in the manuscript are highlighted in yellow in the clean manuscript.\nREVIEWER COMMENTS\nReviewer #2 (Remarks to the Author):\nReview report for “Role of Critical Stress in Quantifying the Magnitude of Fluid-Injection Triggered Earthquakes” by Yu et al.\nOver the past decade or so, fluid-injection-related industrial operations have been performed for a variety of purposes, including wastewater disposal, hydraulic fracturing, enhanced geothermal. As a byproduct, these operations have also caused small-to-moderate earthquakes, sometimes in a relatively stable region and with an unexpected magnitude, thus raising serious concerns on regulation, safety issue, mental health, etc. As such, it is important to know the proper protocols or strategies during fluid injection operation, for example, how to prevent the occurrence of un-controlled, potentially runaway, earthquakes. To this end, Yu et al. have conducted a series of triaxial experiments, by injecting homogeneously-distributed fluid into a fully saturated and prestressed fault. They focus on the role of prestress in modulating the relation between the cumulative (or the maximum) seismic moment and the volume of injected fluid. They find that increasing the proximity of fault’s prestress to failure can reduce the amount of injected fluid required for fault reactivation. Equivalently, injecting the same amount of fluid into a fault with higher susceptibility to failure tends to produce a larger event. These findings hold true for two different types of loading conditions: constant shear stress (CSS) and zero displacement (ZD). Then, the authors propose a theoretical model for explaining their experimental observations and compare the results with other observations in the lab or in situ. Overall, this study provides a better estimation for the maximum moment of induced earthquakes than the conventional model of McGarr (JGR, 1976, 2014) with a fixed prestress level.\nOn one hand, I find this study interesting and suitable for publication in Nature Communications, as it presents both experimental results and theoretical analysis, which can have a direct and timely impact to the scientific and industrial communities. On the other hand, I am afraid the novelty of this study may be compromised, at least partly, by a recent study from the same research group (Li et al., Nature Communications, 2021, doi:10.1038/s41467-020-20700-4). If the authors can provide more evidence to show that this study includes significantly new understanding on the role of prestress, then I will be happy to make my further recommendation. Below I list my detailed concern on novelty as well as some other comments.\nMajor comments:\n(1) Comparison to previous studies\nSeveral previous studies have already shown that factors other than the total volume of injected fluid can influence the nature of fluid-injection-induced earthquakes, e.g., whether they are confined to the zone of pressurization or can expand beyond it. These factors include normal stress, initial shear stress, injection rate, and aseismic slip (e.g., Galis et al., Science Advances, 2017, doi:10.1126/sciadv.aap7528; Passelègue et al., GRL, 2018, doi:10.1029/2018GL080470; Bhattacharya and Viesca, Science, 2019, doi:10.1126/science.aaw7354; Ji and Wu, Tectonophysics, 2020, doi:10.1016/j.tecto.2020.228572). I think these studies should be appreciated in the introduction to tell readers the current stage of understanding fluid-injection-induced earthquakes. Especially, the work of Li et al. (Nature Communications, 2021, doi:10.1038/s41467-020-20700-4) should be discussed, not only because it was done by the same research group and published in the same journal, but also because it also investigated the role of prestress and proposed a theoretical model very similar to this study. As such, I feel the introduction part, e.g., lines 44-57, needs to be revised. The revised version should tell readers what have already been done (specifically on the role of prestress) and set up the stage for this study (e.g., what else can be done for improving the understanding of the role of prestress).\nAgreed. We have expanded this narrative to discuss the current state of characterizations and observations in field and laboratory, controlling factors and to define key remaining uncertainties in relating to field observations. Please see added/revised text.\n(2) The uniqueness of this study\nFollowing (1) and regarding the relation to Li et al. (Nature Communications, 2021, doi:10.1038/s41467-020-20700-4), I think the authors should do a better job advertising the unique aspects (or findings) of their study. From my own reading, their experimental configuration (triaxial) is not exactly the same as that in Li et al. (double direct shear). Consequently, the results could be different. For example, during the CSS test of this study, the maintenance of constant shear stress implies a continued increase of differential stress (\\sigma_1 - \\sigma_3) to compensate the shear stress release due to slip. But this also means there is an increase of normal stress (the term after + in Eq. 1) to compensate the normal stress reduction due to fluid injection. This situation is different from that under a double direct shear in Li et al. (NC, 2021), where normal stress change is solely caused by fluid injection. Moreover, the temporal evolution of injection-induced displacement seems to be different: it shows decreasing slope under CSS and increasing slope under ZD in this study (Fig. 3), but shows increasing slope under CSS and decreasing slope under ZD in Li et al. (their figure 1). I am not sure if such contrasting results can be attributed to the difference in experimental configuration, but I do encourage the authors to think about the new findings of their study.\nAgreed. We now clarify and emphasize the unique contribution of the study. In the revised text we have defined key remaining issues in the role of pre-stress in defining the precise form of the relationship – in particular we (i) define the precise form of the prefactor ( ) that elevates the mome𝛽nt 𝑀 =𝛼𝐺Δ𝑉 and (ii) rationalize the selection of the gradient term ( as unity rather than 3/2. Current field 𝛼 observations for discrete magnitudes of triggered events leave equivocal the range of gradients 𝛽) that honor data (see Galis et al., 2017) - a serious shortfall as =3/2 would significantly elevate 1< 𝛽 < magnitudes for larger injected volumes. We make the case that =1 – congruent with observations of field 3/2 𝛽 projects (Bentz et al., 2020), and include a new figure (Fig. 5(b)) to emphasize this point. Finally, we (iii) 𝛽 reproduce full sequences of in the laboratory that demonstrate a new sigmoidal form that is both 𝑀−Δ𝑉 consistent with analysis and supported by field data. Please see added/revised text.\nThe principal distinction between Li et al. 2021 and this paper is in the careful and systematic observation of the response over the full phase space as first-of-their-kind laboratory observations that parallel the form of field observations (Bentz et al., 2020) and support the gradient of as 1:1 𝑀−Δ𝑉 rather than 1:3/2.\n𝑀−Δ𝑉\nAnd yes, as you note, the experimental geometries are different. We use the single-inclined-fracture (SIF) triaxial geometry as it allows simultaneous measurements of end-to-end acoustics (not reported in this paper), better control of transient pore pressures in isolating them from total stress effects (not reported in this paper) – but principally to allow acoustics. And as you note, linked changes in shear and normal stresses, and evolving fault contact areas are additional complications over DDS but these corrections are readily applied. We do not interpret any slight differences in slopes in our observations as systematically influenced by the DDS versus SIF geometries – we interpret this as normal variability 𝑀−Δ𝑉 among experimental runs.\n(3) The estimation of shear displacement (lines 84-85)\nI wonder if the authors have made corrections for the elastic deformation of the rock sample and that of the loading machine. In principle, to estimate the true shear displacement along the fault, elastic deformation from various sources needs to be subtracted from the raw record of load-point displacement (e.g., see Ji and Wu, Tectonophysics, 2020, doi:10.1016/j.tecto.2020.228572).\nAxial displacement is corrected for elastic shortening through calibrations with steel blank samples. Axial loads are corrected for membrane/jacket resistance. Shear displacements Δu are calculated as the along-fracture component of displacement as , where is the axial displacement of the sample measured by a linear variable differential transducer. Fault compaction/dilation-related Δ𝑢 = Δ𝑢𝑎/cos𝜃 Δ𝑢𝑎 displacements are neglected in this as their magnitudes are small relative to shear offset. Similar simplifications have been used and justified in prior experiments, e.g., Ye and Ghassemi, 2018.\nI also wonder if the authors have made corrections for the change of “actual” contact area during slip, as it is known that in typical triaxial test the “actual” contact area can decrease with slip.\nYes, we have corrected for the actual contact area taking account of the ‘shortening’ due to slip.\n(4) Fig. 4 and lines 192-193\nI had a hard time in understanding the fault behavior during the so-called “first pre-slip stage”. According to the description in lines 192-193, fault is locked at this stage, but Fig. 4 shows that the cumulative seismic moment is not zero during this stage. So, either the authors forgot to make corrections for elastic deformation (see Comment #3), or the tangential stiffness is not a constant (as both shear stress and normal stress are increasing during axial loading). Some clarification is needed: whether there is only elastic deformation or small amount of fault slip during “pre-slip stage”.\nAgreed. Narrative added. This is shear deformation that results from change in effective stress/pore pressure and is very small (Fig. 3) at ~two-orders-of-magnitude smaller than the unlocked reactivated displacement (Fig. 4). These displacements are in micrometers, in contrast to the thousands of micrometers of displacement observed during the \"post-slip\" stage. This observation is a curiosity, but a real effect and measurable response and speaks to the sensitivity of our experiment with regard to these infinitesimal shear displacements and our highly resolved measurements of fluid injection volumes.\nThe point at which this initial 1:1 slope in Fig. 4 transitions to the (near) vertical represents fault reactivation. Before finally assuming a 1:1 trajectory as it slides at steady friction as pore pressures increase linearly with injected volume through a constant fault normal stiffness, defining this sigmoidal shape for as first 1:1 – then vertical – then 1:1. The ultimate segment being parallel to the McGarr line but vertically offset as a function of pre-stress as .\n𝑀−Δ𝑉\n𝑐\n𝑀𝑚𝑎𝑥 = 1−𝑐𝐺Δ𝑉\n(5) Moment magnitude or moment\nIn many places (e.g., line 118, 174, 189, 190, 205, 283, 286, 290, “Event magnitude” in Fig. 3), the authors have incorrectly used “moment magnitude” for “moment (M0)”. Please correct the use.\nAgreed. Thanks. Modified.\n(6) Abnormal results for CSS4 (Table 1)\nCould the authors explain why CSS4 only caused 3760 micrometer cumulative displacement, despite a similar condition to CSS3? Plus, I also notice the stiffness value for CSS4 seems to be abnormal compared to other cases.\nThe inconsistency observed in test CSS4 is real and is attributed to slight variations in fracture surface conditions after repeated shear experiments and this is consistent with the different measured stiffness – no other inconsistencies are apparent . We include this data set as, despite the slight inconsistency in magnitude, test CSS4 demonstrates clear and consistent fluid injection-induced slip behavior that would contribute to the overall illustration of response (see Fig. 3a),\n(7) Assumption of nearly complete stress drop\nIn writing Eq. (9), nearly complete stress drop (from initial shear stress c*\\tau_s to zero residual shear stress) is assumed. Could the authors provide some rationale for supporting such assumption?\nWe now clarify this in the text, and here. Yes, the stress drop, that is modulated by the constant c can be large for injection triggered events. For injection triggered events the stress drop is limited only as the 𝜏𝑠 change in effective stress as and not, as for tectonic earthquakes as (for a near invariant normal stress). Thus, large excess pressures, generated by large injected volumes, may 𝜏𝑠 = 𝜇Δ𝜎𝑛′ 𝜏𝑠 = (𝜇𝑠 −𝜇𝑑)𝜎𝑛′ result in very low effective stresses. This magnitude of the stress drop is controlled by the change in effective stress, that in turn is defined by the injected fluid volume – as similarly modulated and unbounded in McGarr expression as . See figure below.\nΔ𝑉\n𝑀 = 𝜇𝐾Δ𝑉~𝐺ΔV\n(8) Line 241 and Eq. (13): In deriving this equation, a Gutenberg-Richter (G-R) distribution with b value at 1 has to be assumed, as in McGarr (JGR, 2014). But such distribution does not hold for the authors’ case, where the experimentally-generated events seem to follow a rather concentrated distribution (Fig. 3), perhaps related to the generation of characteristic earthquakes(?). I guess the underlying reasons are that events generated in the lab are controlled by machine stiffness and constant increment of fluid pressure (0.1 MPa, line 101), and that the experimental fault is rather uniform (not favorable for producing a G-R distribution with b around 1). I think these differences between laboratory settings and natural conditions should be emphasized in the text. These differences could complicate the comparison between laboratory events and natural events, so that some statements (e.g., lines 306-308) may have to be tuned down a bit.\nAgreed. We now note that the cumulative seismic moment is exact and conforms well in form to field data (Bentz et al., 2020) but that an approximation is needed to recover the maximum seismic moment via the term with an assumption of . See changes in text.\nIn general, the parameter \"b\" wi𝑏ll =be1 close to 1, although fluctuations (up to 30%) around this typical value are observed. The variations depend on factors such as the catalog used, the estimating method employed, and the magnitude range (Frolich and Davis, 1993; Godano et al., 2014). The observation is that the distribution of seismic moment, with its logarithm proportional to the magnitude, remains very stable in both space and time (Kagan 1999; Godano and Pingue 2000). Laboratory faults are clear idealizations of the complexity in both form and response of natural faults.\nWe now add and note that Eq. (13) is indeed an approximation from the cumulative seismic moment, which is exact.\nMinor comments:\nBoth “triggered” and “induced” (e.g., lines 1, 12, 16, and many other places) have been used in this study. I know different researchers may have different preferences, but to avoid confusion, it is better to clarify the meanings of “triggered” and “induced”.\nSome researchers distinguish between \"induced\" and \"triggered\" seismicity based on whether the released strain energy is entirely anthropogenic or only partially. They use \"induced\" when most of the energy is anthropogenic and \"triggered\" when it's mostly natural (e.g., McGarr and Simpson, 1997; McGarr et al., 2002). Although there is not a rigorous convention on the naming, since we prefer triggered for the title, and the essence of the paper is that we release the full stored strain energy (from pre-stress) once the event is triggered, we change to triggered throughout, for clarity. See change to text.\nLine 25: please define G (shear modulus)\nAgreed. Defined.\nLines 27-28: perhaps add “some” before “triggered earthquakes”? Also, the authors use “EGS sites” here. I guess they want to refer to the famous Pohang earthquake case, but are other cases (earthquakes with larger magnitudes than expected) also at EGS sites? If not, I would suggest removing “EGS sites”.\nAgreed and modified. We are referring to the Pohang earthquake case specifically.\nLine 45: add “South Korea” to help understand the location of Pohang.\nAgreed and modified.\nLine 53: strictly speaking, \\alpha is not constant. In Ref. 28, it depends on a normalized prestress parameter c, similar to the current study. In Ref. 29, it also depends on the prestress level \\tau_0.\nOK. Agreed. We now define and “to represent” the pre-stress and modes of rupture. See changes in text.\n𝛼 𝛽\nLine 61: I am not sure if “zero displacement” is the correct terminology. In the context of earthquake science, we usually call it “constant displacement”. I leave it to the authors to decide if they want to make a change.\nAgreed. “Zero displacement” refers to the piston/load-point in our experimental configuration (see Fig. 1) which is locked once it reaches a certain displacement. With the evolving shear displacement measured relative to the locked displacement. We retain the term “zero displacement” to avoid confusion with Li et al., 2021.\nLine 96: please be more explicit, DI = de-ionized\nAgreed. Modified.\nLine 98: in “The axial loading and related shear stress is …”, is should be are\nAgreed. Modified.\nLine 119-120: Eq. (3) may not be obvious to some readers, unless they are familiar with the concepts of transformational strain and stress glut, or the previous work of McGarr. Please add the source for Eq. (3).\nAgreed. Added.\nLines 155-157: the experimental configurations are different between Ref. 28 (double direct shear) and the current study (triaxial), I am not sure if the authors can simply refer to Ref. 28 to support the mentioned statement.\nOK. Agreed. Changed to reflect our true meaning.\nBoth suites of experiments are conducted in a triaxial pressure vessel, with the only difference in the fault configuration (DDS vs SIF). Indeed, reference (28) uses a DDS configuration where a center platen is sheared between two side platens, while we use an SIF configuration where fracture surfaces are sheared against each other. However, and importantly, both studies measure the cumulative injection volume via indirect determination through normal stiffness, that is , where is the area of 𝐴𝜎′𝑛 𝐴∆𝑝 the fault, is the fracture aperture and is the chang∆e𝑉 in= p𝐴or𝛥e𝑎 pr=ess𝑘u𝑛re.= We𝑘 𝑛do this to 𝐴guarantee the resolution of the injection volume measurement ( ) into the small volume fault that is difficult to Δ𝑎 ∆𝑝 resolve from the pump volume, alone – due to the high volume and moderate compressibility of the pump Δ𝑉 fluid, very small volume of the fracture and the large resulting combined volume change that then swamps the fracture signal. It is this technique that we are referring to in citing reference (28) where the fault dilation was also measured with a strain belt (28) to corroborate the fluid volumes. We have Δ𝑉 clarified this focus.\nFig. 3b: I can see a large-amplitude oscillation of fluid pressure around 3200 s, what has happened?\nThe transient pressure amplitude spike is likely an occasional pump instability possibly from a power surge. The pump recovers its stepped pressure and the experiment continues with no apparent effect (no anomalous change in displacement). We chose not to decimate the data, that would have removed it.\nLines 191-192: Fig. 4 is in log-log scale, meaning that any apparent linear relation could actually mean a power-law relation. If the authors want to highlight a linear relation between cumulative moment and fluid volume, it would be better to add some reference line in Fig. 4.\nThanks. Yes. Fig. 4 is intended to showcase the experimental results. We add 1:1 reference lines to the same data in Figs. 5 (c) and (d), and include the field data for cumulative from Bentz et al., 2020 (new Fig. 5(b)) that is a key feature of this paper.\n𝑀−Δ𝑉\nFig. 5a focuses on the relation related to the maximum seismic moment. Please clarify if “Our data: CSS & ZD” refer to the actual data or the estimated relation based on Eq. (13).\n“Our data: CSS & ZD” refer to the laboratory data. We have clarified this in the manuscript “…we plot for c = 10%, 90%, and 99.9% as the dot-dashed black lines in Fig. 5a together with our laboratory data…”\nLine 271: the mentioned pre-stress to strength ratio of 60% to 90% does not match the one in In Fig. 5b&c (10% to 50%).\nWe set the pre-stress to strength ratios at approximately 60% to 90% in our laboratory experiments (see Table. 1). We have re-evaluated the laboratory data using the loading stiffness noted in the earlier manuscript (rather than the unloading stiffnesses noted in the previous manuscript). These change the calculated moments and better match the field data. This change is justified in that the constant shear stress experiments (Fig 5c) are indeed in loading throughout the slip cycle.\nLines 279-282: it is fine to suggest that shale may have a lower shear modulus; but in the authors’ case Westerly granite is used, whose shear modulus is not anomalously low.\nAgreed. We remove the extraneous text and match laboratory measured moments and volumes to field predicted moments and volumes for the same conditions as in McGarr (2014) as G=24 GPa.\nReviewer #3 (Remarks to the Author):\nThis study reports shear reactivation experiments on laboratory faults pre-loaded close to failure and reactivated by the injection of fluid into the fault. Similar experiments have been reported in many papers by the authors and other researchers, particularly Li et al, NC, (2021).\nOnly 2 zero-displacement experiments with a narrow range of critical stress (85% and 89%) were conducted. The relevant data points are far below the scaling lines in Fig. 5c, in comparison to the results in Fig. 5b, showing insignificant seismic risks related to the zero-displacement conditions. More zero-displacement experiments with a wide range of critical stress may need to show the significance of this condition.\nWe use the ZD experiments to demonstrate that we recover the same form in response (3-stage sigmoidal), and with a 1:1 slope on the final portion, and that the moment scales with pre-stress magnitude – all identical to the CSS suite. But with smaller displacements and more difficult to resolve for the ZD experiments, the seismic moments are lower. Certainly, we can complete more ZD experiments. Although the results are unlikely to change the message of the paper – that we see a sigmoidal response with 1:1 gradients that reflect initial elastic response, reactivation, then steady sliding – with vertical offsets scaling with pre-stress and with solely 1:1 rather than 1:3/2 trajectories.\nThe noteworthy outcome is the analytical scaling between cumulative moment magnitude and injected fluid volume with an additional consideration of pre-stress compared to Li et al (2021).\nThe principal distinction between Li et al. 2021 and this paper is in the careful and systematic observation of the response over the full phase space as first-of-their-kind laboratory observations that parallel the form of field observations (Bentz et al., 2020) and support the gradient of as 1:1 𝑀−Δ𝑉 rather than 1:3/2 for cumulative moment versus injected volume, and with offsets scaling with a straightforward model for pre-stress.\nThe cumulative seismic moment incorporating pre-stress is a smart idea and should be the main contribution of this study. However, the validation relying only on experimental data is not fully convincible. The lab-scale experiments cannot accommodate many field-scale seismic processes, such as fault ruptures. So, the validation with induced earthquakes (such as data points in Fig. 5a) is really needed to show the reliability of the scaling. Otherwise, this work is not of significance to the field.\nAgreed, laboratory experiments cannot replicate the myriad variables that influence field-scale seismic processes. What they do allow is precise control on variables (applied stresses, pore pressures and measured displacements) that are difficult to constrain in situ. The form of such sigmoidal 𝑀−Δ𝑉 relationships have not been previously observed in the laboratory and closely follow field observations (Bentz et al., 2020, Fig. 2(a), shown below and in the new Fig. 5(b)) that are uniformly 1:1 in gradient (none are 1:3/2), are vertically separated, and have components that transit to give very large moments for small injection volumes (Pohang) – all features that are observed in experiments and explained with our straightforward model and analysis."}, {"round": 2, "reviewer_comments": {"reviewer_2": "I appreciate the authors’ effort for revising the manuscript. I think most of my comments have been properly addressed. On the other hand, I still have a few additional comments that I hope the authors can consider.\n(1) The abstract\nI suggest re-writing this abstract to deliver a clearer message. For instance, only experienced readers can quickly understand the meaning of the sentences in Lines 24-27. As this work aims to highlight the role of prestress as in the paper title, its effect should be summarized clearly using straightforward sentences. Here I only point out one instance, and I hope the authors can come up with a better version of abstract, which will also help reduce the editor’s work.\n(2) Previously I raised a question on the nature of “first pre-slip stage”. In response, the authors mention that it could reflect “infinitesimal shear displacement”, but otherwise use the term “elastic deformation” in the revised manuscript (Line 19 in the abstract, Lines 185, 304, 336, etc.). While I do suspect the recorded signal may contain some elastic deformation, I also think some inelastic deformation could be involved as well. For instance, it reminds me of the preslip as often observed during the initial stage of rock friction experiments under gradually increased shear stress. Preslip is more precisely known as preseismic slip or premonitory slip, in the sense that the slip falls into the aseismic frequency band and hence is not directly measurable to seismic monitoring (such slip itself, although quite small and slow, still represents inelastic deformation). I am not sure if it is also the case in fluid-injection experiments, but I encourage the authors to think about other possibilities (the current wording “elastic deformation” during the first stage may sound too absolute). Another possible cause could be due to the unbalanced fluid pressure at the initial stage of fluid injection, but this is very difficult for me to judge.\n(3) Compared to the theoretical estimation in Li et al. (NC, 2021), one major difference is whether the normalized prestress parameter “c” also appears in the numerator (Eqs. 12 and 13), which should be testable under low values of “c”. In the authors’ case, M (triggered seismic moment) should be very low or even near zero, but according to Li et al. (NC, 2021), M is finite. I think this difference should also be mentioned, as one difference from Li et al. (NC, 2021).", "reviewer_3": "I am satisfied with the authors' clarifications and answers. After a final read, I believe this manuscript is ready for publication."}, "author_response": "Dear Reviewers,\nWe appreciate the positive comments from the reviewers and the time that they have invested in improving our manuscript. Please see below our itemized responses with modifications applied separately to the marked-up manuscript.\nSpecifically, the revisions have:\n Rewritten the abstract.\n Clarified some concepts.\n Discussed the assumptions present in different models linking seismic moment to pre-stress.\nThe following identifies responses to the review comments. Review comments are shown in blue and responses in red. Changes in the manuscript are highlighted in yellow in the clean manuscript.\nREVIEWER COMMENTS\nReviewer #2 (Remarks to the Author):\nI appreciate the authors’ effort for revising the manuscript. I think most of my comments have been properly addressed. On the other hand, I still have a few additional comments that I hope the authors can consider.\n(1) The abstract\nI suggest re-writing this abstract to deliver a clearer message. For instance, only experienced readers can quickly understand the meaning of the sentences in Lines 24-27. As this work aims to highlight the role of prestress as in the paper title, its effect should be summarized clearly using straightforward sentences. Here I only point out one instance, and I hope the authors can come up with a better version of abstract, which will also help reduce the editor’s work.\nOK. We have revised the content of the Abstract and trimmed it to 150 words.\n(2) Previously I raised a question on the nature of “first pre-slip stage”. In response, the authors mention that it could reflect “infinitesimal shear displacement”, but otherwise use the term “elastic deformation” in the revised manuscript (Line 19 in the abstract, Lines 185, 304, 336, etc.). While I do suspect the recorded signal may contain some elastic deformation, I also think some inelastic deformation could be involved as well. For instance, it reminds me of the preslip as often observed during the initial stage of rock friction experiments under gradually increased shear stress. Preslip is more precisely known as preseismic slip or premonitory slip, in the sense that the slip falls into the aseismic frequency band and hence is not directly measurable to seismic monitoring (such slip itself, although quite small and slow, still represents inelastic deformation). I am not sure if it is also the case in fluid-injection experiments, but I encourage the authors to think about other possibilities (the current wording “elastic deformation” during the first stage may sound too absolute). Another possible cause could be due to the unbalanced fluid pressure at the initial stage of fluid injection, but this is very difficult for me to judge.\nAgreed. The exact mechanism for the first pre-slip stage is unconfirmed. Therefore, we have replaced the term “elastic deformation” with “infinitesimal shear displacement” in the manuscript and listed the possibilities including both elastic deformation and inelastic deformation (e.g., local micro-fracturing).\n(3) Compared to the theoretical estimation in Li et al. (NC, 2021), one major difference is whether the normalized prestress parameter “c” also appears in the numerator (Eqs. 12 and 13), which should be testable under low values of “c”. In the authors’ case, M (triggered seismic moment) should be very low or even near zero, but according to Li et al. (NC, 2021), M is finite. I think this difference should also be mentioned, as one difference from Li et al. (NC, 2021).\nYes. Agreed. We clarify differences between McGarr (2014), Li et al., (2021) and this study in the text to identify the different assumptions in the models and the different bounds on solutions.\nReviewer #3 (Remarks to the Author):\nI am satisfied with the authors' clarifications and answers. After a final read, I believe this manuscript is ready for publication.\nThanks."}]

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