The GR-MFD is a relationship that describes the amount and frequency of earthquake magnitudes: N=10a10-bM. The a-value describes the scaling to the total amount of earthquakes N, while the b-value is the proportionality of big-to-small events. Given a catalogue of earthquake magnitudes M, this relationship can be fit through Maximum Likelihood Estimators (Marzocchi and Sandri, 2003). For real datasets, a lower bound truncation called the magnitude-of-completeness (Mc) is introduced to account for detection incompleteness. Many methods exist to evaluate Mc robustly, to account for the incompleteness in the detection of low magnitude events, thusly to avoid biases during the GR-MFD fitting process. In this study, Mc is selected by searching for the value that maximizes the goodness-of-fit metrics like R2 (Schultz et al., 2018) or minimizes the negative log-likelihood of the GR-MFD. In cases with a goodness-of-fit plateaus/valleys, we select the value of Mc closest to plateau start. Sometimes we are conservative in our Mc choice by selecting a value slightly larger than optimal. Note that this conservativism will have a detrimental effect on finding bound cases. The Mc value selected by this process is typically near (but skewed right-ward of) the peak bin of the non-cumulative GR-MFD.

As part of the simple indicators of MMAX, a visual comparison of the GR-MFD fit against the observed data is used to qualitatively examine for the presence of MMAX. Catalogues that are relatively deficient in large magnitudes (compared to their fits) are possible candidates for an MMAX. We note that this deficiency in large magnitude events is a key metric for identifying if/when MMAX can be constrained (Holschneider et al., 2011; Schultz, 2024, 2026a).

To provide additional semi-quantitative assessments of an MMAX, we examine the empirical degree-of-truncation (δMLRG), which is the discrepancy between the largest observed event minus the largest expected event. The expected MLRG can be estimated: if a GR-MFD is assumed, then order statistics suggests a modal value of MLRG=Mc+log⁡10(N)/b (van der Elst et al., 2016). Where N is the total number of events larger than the magnitude-of-completeness Mc and b is the previously described b-value. We can use the prior GR-MFD fits to determine the expected value of MLRG and then compare that against the observed value (i.e., δMLRG). We also use the inverted cumulative distribution function (van der Elst et al., 2016) to determine the percentile of the δMLRG discrepancy – or the likelihood of this degree-of-truncation occurring, assuming an unbound catalogue.

As part of the simple indicators of MMAX, catalogues that exhibit negative δMLRG suggest the presence of an MMAX. We note that the δMLRG discrepancy is a proxy metric for the resolvability of CAP-tests. Usually values of -0.5M (or less) for MLRG-MMAX are required to confidently assert the presence of an MMAX (Schultz, 2024, 2026a). Note that this MLRG-MMAX difference is the expected value where quantitative inferences of MMAX can start being made with 95 % confidence, from theoretical considerations (Eq. 15; Holschneider et al., 2011).

The first test is rooted in the statistical framework of hypothesis testing. Here, we take advantage of the fact that the distribution of magnitudes (M) and the distribution of jumps in the sequence of large events (ΔMLRG) is the same if unbound, but differ when there is an MMAX upper bound (Schultz, 2024, 2026a). This fact ideally lends itself to hypothesis testing via the KS-test (Berger and Zhou, 2014). Given a catalogue, both M and ΔMLRG can be observed. We can then compare these two observations against each other to test if they are drawn from the same distribution (or not), via the KS-test. This approach is advantageous in that it is non-parametric – i.e., it is completely data-driven and imposes no assumptions about the kind of distributions M or ΔMLRG were drawn from. Because of this, we do not need to fit the data to a GR-MFD or have any knowledge/estimates of the b-value to perform our KS-test. Confidence in the KS-test is reported as compliments of standard p-values, where 95 % is a common threshold used to declare statistical significance.

Since this test is only interested in discerning the existence of an MMAX, additional catalogue realizations can be drawn through reshuffling the order of events. In this sense, bootstrapping can be employed to generate numerous catalogue realizations in which the KS-test is repeated. This provides more robust p-value estimates. Testing on both synthetic and real datasets suggests that the KS-test is significantly more sensitive to discerning MMAX than approaches that attempt to appraise GR-MFD fits (Schultz, 2024, 2026a). As well, synthetic testing on unbound cases shows that this KS-test produces false-positives at the rate expected for a p-value distribution (Schultz, 2024). To be able to discern the influence of MMAX, MLRG-MMAX discrepancies of -0.5M or better are usually required, consistent with theoretical expectation (Holschneider et al., 2011).

While this formulation of the KS-test is powerful, we also provide a word of caution towards a potential interpretation pitfall: this method is testing for differences between the distributions M and ΔMLRG. The presence of an MMAX is one possible reason for this difference, but others may also confound a clear interpretation (e.g., temporal changes in b-value, kinked distributions, tapered distributions). Thus, the KS-test should be suitably pre-processed or complemented with other tests to increase the certainty of an MMAX interpretation.

The next test is rooted in the statistical framework of Maximum Likelihood Estimation. If there is some suggestive evidence for the existence of an MMAX, the next natural step is to quantify this value. The log-likelihood function is defined as follows: 1lnLM;θ=∑iln⁡fMM;b,Mc,MMAX-∑jln⁡(fM(ΔMLRG;b,0,MMAX-MLRG)) where the probability density function of the GR-MFD is given by fM(M), with a set of model parameters θ (Schultz, 2024). This function essentially entails two parts: the log-likelihood for the catalogue magnitudes M (right-hand side of equation, first term) and the log-likelihood of the jumps in largest events ΔMLRG (right-hand side of equation, second term). The optimal set of model parameters θ are then solved for via numerical methods to maximize the log-likelihood, given the observed catalogue data M and ΔMLRG. We perform this optimization in two-steps: the first using the standard log-likelihood constrain the b-value (Marzocchi and Sandri, 2003) (with the optimal Mc estimates) and then using the composite log-likelihood to constrain MMAX. In this study, we consider the simplest MMAX variant for fM(M), which is a truncated GR-MFD (Schultz, 2024).

If this test is only interested in discerning a stationary value of MMAX, additional catalogue realizations can be drawn through reshuffling the order of events. Similar to the KS-test, bootstrapping can be employed to generate numerous catalogue realizations in which the MLE-test is repeated. This provides more robust MMAX estimates. Testing on both synthetic and real datasets suggests that the MLE-test is sensitive to quantifying MMAX within a hundredth of a magnitude unit when M_LRG-MMAX discrepancies are -0.5M or better. In cases where the MLE-test is applied to unbound catalogues, bootstrapped estimates of MMAX will be much larger than MLRG and standard deviations can be on the order of 1 magnitude unit or greater.

The third and final CAP-test is rooted in the statistical framework of likelihood inference. Together, the two prior tests provide suitable evidence for the existence of MMAX. However, MMAX may be a function of time or injected volume, becoming some non-stationary value throughout the catalogue duration. Certainly, MMAX processes relevant for induced seismicity and hydraulic fracturing have been proposed in the past (McGarr, 2014; Hallo et al., 2014; Galis et al., 2017; Elsworth et al., 2025; Im and Avouac, 2025; Sáez et al., 2025). In this sense, having an approach that can distinguish the best proposed MMAX model (given the data) would be insightful.

The EW-test starts by considering an ensemble of proposed MMAX models to explain the catalogue data (Schultz, 2024). Using the previously defined log-likelihood function, both Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) can be defined for each MMAX model (Schwarz, 1978; Akaike, 1998); note that we have used the small sample size corrections for AIC/BIC (Sugiura, 1978; McQuarrie, 1999). Next, we compute the differences in AIC/BIC scores, by subtracting the score of the best MMAX model. These score differences can be translated into relative model weights by an exponential function (Wagenmakers and Farrell, 2004). We combine AIC/BIC weights into a single weight by taking an average between the two. In this study we will consider four standard MMAX models: McGarr-like (i.e., log-proportional to injected volume; MMAX∝log⁡10V1) (McGarr, 2014; Hallo et al., 2014; Elsworth et al., 2025), Galis-like (i.e., MMAX∝log⁡10V3/2) (Galis et al., 2017), a constant tectonic upper bound (Kanamori and Anderson, 1975), and the unbound null hypothesis. The unbound null hypothesis consists of three model parameters (Mc, b-value, GR-MFD variance), while all bound models have one additional model parameter (i.e., K=4) to account for the slope of the volume-based relationship.

The interpretation of the MMAX model weights is straightforward: larger model weights indicate a better explanation of the data. The model with the largest weight is the best explanation of the data (within the ensemble). To quantify the statistical significance of weight differences between two models, the relative odds ratio can be computed as the ratio of the two model weights (larger/smaller). Ratios of 1+ are insignificant, 3+ are substantial/positive, 10+ are strong, and 100+ is decisive (Kass and Raftery, 1995). We note that the best model in an ensemble does not necessarily imply the veracity of the model; there could be another (unknown) model that explains the data better than all of those yet considered.

In synthetic testing, the EW-test can accurately and confidently discern the true MMAX model (Schultz, 2024). Usually, only a handful of ΔMLRG observations are required to confidently identify the true model (i.e., with odds ratios of 3–10 or better). However, like the prior CAP-tests, MMAX must be influencing the catalogue for meaningful inferences to be made. Said another way, if the MMAX is much larger than MLRG, then the EW-test will not be able to distinguish between bound/unbound hypotheses. A schematic diagram explaining the EW-test has been provided (Fig. 2).

The Äspö HRL is located on the Baltic east coast of Sweden, near the Simpevarp Peninsula (Fig. 3). This UGL was constructed as a testbed to study the potential for nuclear waste storage by the Swedish Nuclear Fuel and Waste Management Company starting in 1986. By 1995, construction efforts reached the 450 m target depth into the granitoids of the Trans-Scandinavian Igneous Belt (Stanfors et al., 1999).

The intention of recent (June 2015) injection experiments at the Äspö HRL was to test the feasibility of cyclic stimulation as an alternative means to enhance subsurface permeability, while also reducing the severity of induced seis micity (Zang et al., 2019, 2021). Six injection stages (HF1–HF6) were conducted in a single borehole (28 m long at 410 m depth), with an average stage spacing of ∼ 3 m, and encountering either Ävrö granodiorite or fine-grained diorite-gabbro. Stage locations were chosen to avoid natural fractures (Zimmerman et al., 2019). Each stage used between 4.1-27.2 L of injected fluid, spread between 4–6 injection sub-cycles (Zang et al., 2019). Some stages (HF1, HF2, HF4, and HF6) used traditional hydraulic fracturing techniques, while other stages (HF3 and HF5) employed cyclic stimulation (Zang et al., 2019). Resulting microseismicity was predominantly recorded in stages HF1-HF2 (López-Comino et al., 2017; Niemz et al., 2020, 2021), with events reaching up to Mw-3.5 (Kwiatek et al., 2018) and slipping with reverse or strike-slip motions (López-Comino et al., 2021). Cyclic stimulations tended to produce less seismicity (Zang et al., 2019) and more complex fracture networks (Stephansson et al., 2019; Zhuang et al., 2019), albeit with less permeability enhancement (Zimmerman et al., 2019). Data for the Äspö HRL is publicly available (Zang et al., 2024) and a spatiotemporal summary of stimulation events is plotted (Fig. 4).

To begin assessing if some process might be restricting magnitude growth at the Äspö HRL, we fit the GR-MFD to the catalogue data from each injection stage (i.e., HF1-HF6). To account for magnitude errors, we employ a 50-trial bootstrap process in which the catalogue magnitudes are dithered by ±0.1 (this includes a dithered Mc). Only four of the stages (HF1-HF3, and HF6) have enough data to examine. Overall, the earthquakes here appear to be deficient in large-magnitude events (Fig. 5). Correspondingly, the δMLRG discrepancies range between -0.2 to -0.4 for all stages. Assuming an unbound catalogue, these degrees-of-truncation (δMLRG) would be < 1st, 3rd, < 1st, and 6th percentile events, respectively. These initial assessments are suggestive of some MMAX upper bound restricting catalogue growth at the Äspö HRL.

Next, we use the CAP-tests to detect and assess the potential for MMAX more rigorously. The KS-test is performed 50 times, in which the catalogue magnitudes are dithered (including a dithered truncation magnitude). The KS-test also performs 100 reshuffles within each trial. Average KS-test confidences are > 99.99 %, > 99.99 %, 99.87 %, and 68.85 % for stages HF1-HF3 and HF6, respectively. We perform 50 MLE-tests using dithered catalogues and 100 reshuffles within each trial. Similarly, MLE-tests suggest the standard error in fitted MMAX values is 0.02, < 0.01, 0.02, and 0.18 for stages HF1-HF3 and HF6, respectively. Next, we seek test hypotheses for the functional form of MMAX: our EW-tests focus on McGarr-like (McGarr, 2014; Hallo et al., 2014; Elsworth, et al., 2025), the Galis-like model (Galis et al., 2017), a constant tectonic upper bound (Kanamori and Anderson, 1975), and the unbound null hypothesis. Similarly, EW-tests also show evidence for an MMAX bound process, most strongly evidenced with stage HF2 (Figs. 6 and S1–S3). By the end of HF2 stimulation, the Galis-like model is > 100 times more likely than the unbound model (Fig. 6).

Last, we organize our results for all injection stages at the Äspö HRL – for convenience to the reader. These results are summarized below (Table 1).

All the stages at the Äspö HRL appear to have some evidence of a bound process restricting the growth of earthquake magnitudes. The most convincing of the cases is also the most well-resolved one: for HF2, all the simple/CAP-tests unanimously indicate a bound process with strong statistical significance. As the cases become less well-resolved, the statistical confidence also diminishes. For example, the HF1 and HF3 stages have most CAP-tests indicating a bound process, only the EW-test of HF3 narrowly falls short of statistical significance. The HF6 stage produces the most mixed results: simple-tests, MLE-test, and EW-test are indeterminate, while the KS-test is unbound. It is worthwhile to mention that HF6 differed in stimulation approach in two ways: the first sub-stage of HF6 was the largest injected volume (with reduced volume for later sub-stages) and the last three sub-stages were performed on a different day.

Overall, we interpret stage HF2 as certainly being bound. Stages HF1 and HF3 as likely to be bound, but likely needing more well-resolved catalogues to better discern MMAX. We hesitate to make a clear statement for HF6, which appears indeterminate due to data limitations.

The interpretation that all the Äspö HRL stages were (likely) bound corresponds with the geophysical interpretations from relevant studies. The intention of the Äspö HRL was to test various stimulation programs against the complexity/growth of hydraulic fractures (Zang et al., 2019, 2021). Stage intervals were chosen to avoid natural fractures, and impression packers noted the generation of new hydraulic fractures (Zimmerman et al., 2019). Furthermore, the progressive growth of the HF2 hydraulic fracture plane was inferred jointly from the microseismic and deformation constraints (Niemz et al., 2020, 2021). In this sense, a progressively growing fracture aligns well with the bound interpretation: the finite extent of a fracture limits MMAX via geometric considerations (Kanamori and Anderson, 1975). As stimulation continues, the fracture continues to grow; thus, the value of MMAX would increase alongside the injected volume. Correspondingly, each of the Äspö stages indicated bound growth (albeit with varying degrees of confidence).

These interpretations at the Äspö HRL constitute the simplest interpretation. We describe a scenario as to how hydraulic fracturing would be linked to a bound MMAX interpretation. Regardless of the fracture network's complexity, the finite spatial extent of stimulated fractures ultimately restricts magnitude growth.

The SURF is located near Lead, South Dakota, and has repurposed the Homestake gold mine (Fig. 3). The SURF is a research facility operated by the South Dakota Science and Technology Authority to study rare-process physics (Heise, 2015). The EGS Collab takes advantage of the SURF, by using this facility to host hydraulic stimulation experiments at field-scale depths of ∼ 1.5 km (Dobson et a., 2018; Kneafsey et al., 2018; Morris et al., 2018).

The EGS Collab Experiment #1 intended to connect injection and production boreholes via stimulated fractures in a controlled environment (Kneafsey et al., 2020; Morris et al., 2018). Injection and production boreholes were drilled subparallel to the estimated minimum principal stress direction (Oldenburg et al., 2017) in the phyllites of the Precambrian-aged Poorman Formation (Kneafsey et al., 2020). These metamorphic rocks are strongly foliated and highly anisotropic (Frash et al., 2019; Vigilante et al., 2017). The rock mass is generally thought to be low permeability (∼10-18 m²), with a thermally altered stress field (Singh et al., 2019). Some natural fractures were noted in borehole cores (Fu et al., 2021), with at least one noteworthy fracture that is naturally/actively flowing (Wu et al., 2021a). Starting in May 2018, stimulation stages took place at three wellbore intervals (N164, N142, N128), covering both hydraulic stimulation and hydraulic characterization programs; stimulation programs injected on the order of 10 s L of water per interval (Morris et al., 2018; White et al., 2019). Resulting microseismicity was recorded for each interval, with ∼ 2000 located events (Schoenball et al., 2019, 2020; Chai et al., 2020; Qin et al., 2024). Studies have covered topics including modeling heat transport (Wu et al., 2021b), strain/deformation (Guglielmi et al., 2021), and hydraulic fracture propagation (Li and Zhang, 2023). Data for the EGS Collab Experiment #1 is publicly available (https://gdr.openei.org/, last access: July 2025) and a spatiotemporal summary of stimulation events is plotted (Fig. 7).

To begin assessing if some process might be restricting magnitude growth at SURF Experiment #1, we fit the GR-MFD to the catalogue data from each injection stage (i.e., N164, N142, N128). To account for magnitude errors, we employ a 50-trial bootstrap process in which the catalogue magnitudes are dithered by ± 0.1 (this includes a dithered Mc). Only one of the stages (N164) appear to be appreciably deficient in large magnitude events (Fig. 8). Correspondingly, the δMLRG discrepancy is significant for N164 (-1.15M), but fairly small for N142 (-0.05M) and N128 (-0.28M). Assuming an unbound catalogue, the δMLRG discrepancy for N164 would be a < 1st percentile event; stages N142 (10th percentile) and N128 (33rd percentile) are more commonly expected occurrences. These initial assessments are suggestive of some MMAX upper bound restricting catalogue growth for one stage at SURF Experiment #1.

Next, we use the CAP-tests to detect and assess the potential for MMAX more rigorously. The KS-test is performed 50 times, in which the catalogue magnitudes are dithered (including a dithered truncation magnitude). The KS-test also performs 100 reshuffles within each trial. Again, KS-test results are split, with average confidences of 99.96 % for stage N164, but 34.23 % and 74.39 % for stages N142 and N128, respectively. We perform 50 MLE-tests using dithered catalogues and 100 reshuffles within each trial. MLE-tests are also split: standard error in fitted MMAX values are 0.10, 2.18, and 1.31 for stages N164, N142, and N128, respectively. Similarly, EW-tests also shows suggestive evidence for an MMAX bound process with N164 (Fig. 9), but certainly not for stages N142 and N128 (Figs. S4–S5 in the Supplement).

Last, we organize our results for all injection stages at SURF Experiment #1 – for convenience to the reader. These results are summarized below (Table 2).

CAP-test results at SURF Experiment #1 have similarities and differences from those at the Äspö HRL. For example, stage N164 is similar to the stages at the Äspö HRL: both simple-tests and CAP-tests unanimously agree on an MMAX bound – with varying degrees of confidence. On the other hand, stages N142 and N128 differ in that they produce strong and unambiguous unbound inferences. Because of this, we interpret stages N142 and N128 to be truly unbound.

The interest in considering UGL cases is the wealth of complementary geophysical information to cross-examine against the results of CAP-tests. The stimulation at stage N164 is predominantly understood to be the creation of a new fracture network. This interpretation comes from multiple lines of evidence: the orientation of microseismic fault planes with respect to the ambient stress field (Schoenball et al., 2020), deformation constraints on fracture motion (Guglielmi et al., 2021), significant recovery of injected fluid (White et al., 2019), and direct evidence of new fluid jets intersecting the producing well (Fu et al., 2021). The later N164 injections (i.e., June 2018) were performed from the producing well side, into the newly created fracture network. That said, some complexity in the N164 stimulation suggested limited interaction with natural fractures, via either arrests or redirected continuation of fracture growth (Schoenball et al., 2020; Fu et al., 2021). Overall, these inferences are consistent with finite hydraulic fractures bounding the growth of earthquake magnitudes – which are the same interpretations made for the Äspö HRL.

On the other hand, stages N142 and N128 had significant interactions with natural pre-existing fractures, and likely reactivated them in shear rather than creating a new fracture network. These inferences/interpretations come from the orientation of microseismic fault planes with respect to the ambient stress field, corroboration with well image logs, and observations of shear deformation (Schoenball et al., 2020). The connection to fractures/faults allows for seemingly unbound magnitude growth, as the spatial extent of these pre-existing structures have the potential to host larger events.

These interpretations at SURF Experiment #1 constitute the next level of complexity in interpretation. We describe a scenario where hydraulic stimulation could interact with natural fractures/faults to produce an unbound process. Said another way, the interaction with natural fractures/faults (of sufficient size) can facilitate unbound magnitude growth.

The GTS is located near the Grimsel pass in the central Swiss Alps (Fig. 3), ∼ 450 m below the Juchlistock in the Varsican-aged Aar Massif granites (Schneeberger et al., 2019). The GTS was established in 1984 as an underground research facility to study the safe disposal of nuclear waste. It is owned and operated by the National Cooperation for Radioactive Waste Disposal (Nagra; https://www.grimsel.com/, last access: March 2025).

The intention of recent (February–May 2017) injection experiments at the GTS was to demonstrate the stimulation of fractures at the decameter scale and to better understand how to manage induced seismicity (Gischig et al., 2016; Amann et al., 2018). Twelve injection stages spanning ∼ 1 m intervals were situated in two boreholes (∼ 45 m long) (Gischig et al., 2020): six related to stimulation via hydraulic fracturing (HF1-HF8) (Dutler et al., 2019) and six more via hydroshearing (HS1-HS8) (Krietsch et al., 2020b). Each stage used (on the order of) ∼ 1000 L of injected fluid, spread between 4 injection sub-cycles. The role of natural faults and fractures are a prominent focus of the GTS injection experiments, which injected into either brittle-ductile shear zones, ductile shear zones, or intact rock (Doetsch et al., 2018a). The degree of seismic response for individual stages was strongly heterogeneous in space, with stages HS4, HS5, and HF2 being both the most seismically active and hosting the large events (M-3.0, M-2.8, and M-3.9) (Villiger et al., 2020). Studies at the GTS were diverse, covering topics like geological characterization (Krietsch et al., 2018), stress inversion (Krietsch et al., 2019), tomographic velocity changes (Doetsch et al., 2018b; Schopper et al., 2020), permeability changes (Jalali et al., 2018; Brixel et al., 2020a, b), and inferring fracture propagation from hydromechanical response (Dutler et al., 2019; Krietsch et al., 2020a, b). Data for the GTS is publicly available (10.3929/ethz-b-000276170, Doetsch et al., 2018c) and a spatiotemporal summary is plotted (Fig. 10).

To begin assessing if some process might be restricting magnitude growth at the GTS, we fit the GR-MFD to the catalogue data from each injection stage (i.e., HSX & HFX). To account for magnitude errors, we employ a 50-trial bootstrap process in which the catalogue magnitudes are dithered by ±0.1 (this includes a dithered Mc). Note that many of the stages recorded here have too few events for a meaningful analysis; thus, we predominantly focus our discussions to a subset of stages. Only two of the stages (HS4 and HF2) appear to be appreciably deficient in large magnitude events (Fig. 11). Correspondingly, the δMLRG discrepancy is large for HS4 and HF2 (-1.31M and -0.61M), but fairly small for the other viable stages (between -0.19M and +0.08M). Assuming an unbound catalogue, the δMLRG discrepancy for HS4 and HF2 would both be < 1st percentile events; all other stages are commonly expected occurrences. These initial assessments are suggestive of some MMAX upper bound restricting catalogue growth for just two stages at the GTS.

Next, we use the CAP-tests to detect and assess the potential for MMAX more rigorously. The KS-test is performed 50 times, in which the catalogue magnitudes are dithered (including a dithered truncation magnitude). The KS-test also performs 100 reshuffles within each trial. KS-test results are split, with strong confidences of > 99.99 % for stages HS4 and HF2, but unconvincing values for all other stages. We perform 50 MLE-tests using dithered catalogues and 100 reshuffles within each trial. MLE-tests are also split: standard error in fitted MMAX values are < 0.01 and 0.02 for stages HS4 and HF2, but values are large (0.34–1.92) for all other stages. The EW-tests shows decisive evidence for an MMAX bound process with HS4 (Fig. 12), and strong evidence for HF2 (Fig. S6). All other stages have substantial-to-strong evidence for an unbound process via EW-tests (Figs. S7–S10).

Last, we organize our results for all injection stages at the GTS – for convenience to the reader. These results are summarized below (Table 3).

In summary, most of the GTS stages appear to exhibit unbound growth of earthquake magnitudes. Although, this interpretation varies in confidence, depending on the specific stage in question, since many of the stages had relatively few events recorded. The most confidently unbound case is HS5, which fails all of the simple-tests and CAP-tests. On the other hand, there are two exceptions to this general trend: HS4 shows clear and definitive evidence for a bound process, while HF2 shows strong evidence for a bound process. That said, stages with fewer than 50 events are difficult to arrive at a clear interpretation between truly unbound or simply lacking data.

The aims and scope of the GTS were to investigate the response of injection into faults/fractures at the intermediate scale (Gischig et al., 2016; Amann et al., 2018). For example, all HSX stages intentionally injected into previously known fractures of shear damage zones (Krietsch et al., 2018); thus, micro-seismically delineated fault planes were generally consistent with fracture/fault orientations (Villiger et al., 2020), near-field hydromechanical effects were consistent with pre-existing fracture dislocation (Krietsch et al., 2020b), and many stages were hydraulically connected to other boreholes via these stimulated fractures (Brixel et al., 2020b). In particular to HS5, seismicity was well fit to a single fault plane (∼ 16 m diameter) oriented subparallel to the targeted fracture (Villiger et al., 2020), significant pressure increases (∼ 70 %–75 % of injection pressure) were observed in boreholes 7–8 m away (Krietsch et al., 2020a), likely due to transient permeability increases driven by fracture aperture changes (Krietsch et al., 2020b). Furthermore, the diameter of the HS5 fault plane (∼ 16 m) is appreciably larger than the largest event diameter (∼ 0.65 m), assuming a circular crack. HFX stages started hydraulically fracturing intact rock and subsequently propagated/stimulated new fractures (Dutler et al., 2019). However, the newly propagating fractures were inferred to have significant interactions with pre-existing fractures, which served as pressure sinks that arrested further growth (Dutler et al., 2019; Villiger et al., 2020). Note that HF6 is unique, since it injected directly into a pre-existing fracture (by mistake) and thus can be considered a hydroshearing experiment. Furthermore, the propagation of microseismic events along pipe-like geometries was thought to be formed via the intersection with natural fractures (Dutler et al., 2019). Given the prior interpretations of CAP-test results at SURF Experiment #1, we would anticipate that all GTS stimulations should be unbound – because of their prominent connection to pre-existing fractures. Bound cases would require special exceptions to this generalization.

On the other hand, the HS4 stage is a clear exception to this generalization. The HS4 seismicity is not well-fit by a single plane. Instead, it is best fit by four intersecting planes (C1–C4), where C1–C3 are oriented subparallel with pre-existing fractures and C4 is likely a new tensile failure (Villiger et al., 2020, 2021). Clustering in focal mechanism slip style also corresponds to spatial clusters (Villiger et al., 2021). Each cluster has seismicity spatially restricted along discrete linear streaks (Villiger et al., 2021); these streaks grow/propagate alongside injection, although their spatial extent (∼ 1–2 m) is much smaller than other HSX fault planes. In tectonic contexts, streaks are often interpretated as rheological boundaries between seismic and creeping/locked fault segments (Rubin et al., 1999; Waldhauser et al., 2004). In other hydraulic fracturing contexts, streaks have also been observed and analogously interpreted (Rutledge et al., 2004; Evans et al., 2005). Thus, the streaks and clustering at HS4 have been interpreted as fractures channelizing fluid-flow towards highly seismogenic asperities that slip perpendicular to the fluid migration direction (Villiger et al., 2021). In this sense, we argue that HS4 seismicity is spatially bound to these asperities/streaks, giving rise to the bound growth of magnitudes. Correspondingly, the GR-MFD for HS4 (Fig. 11a) starts to roll-off around M-4.3 and M-3.0 was the final MLRG value; note that M-3.0 roughly corresponds to a circular crack diameter of ∼ 0.5–1.0 m, which is comparable to the spatial extent of the streaks (∼ 1–2 m).

Similarly, HF2 is also an exception to this generalization. The HF2 seismicity was best fit by two intersecting planes, the first which resembles a newly created hydraulic fracture and then a deflection to a more E–W orientation (Dutler et al., 2019; Villiger et al., 2020). It has been suggested that this stage is exceptional in that it potentially has limited interaction with pre-existing structures, being able to propagate fracture growth before leak-off into potential fracture connections (Dutler et al., 2019; Villiger et al., 2020). In fact, the growing hydraulic fracture intersected monitoring boreholes, reaching further than suggested by the microseismic events (Dutler et al., 2019). In this sense, we would argue that HF2 is most similar to the N164 stage at SURF Experiment #1. Correspondingly, both HF2 and N164 express a bound process via CAP-tests.

These interpretations at the GTS constitute the greatest level of interpretive complexity we will consider in this study. We have reiterated scenarios where hydraulic stimulation could interact with natural fractures/faults to produce an unbound process. Regardless of the stimulation program, the interaction with natural fractures/faults (of sufficient size) can facilitate unbound magnitude growth. Exceptions come in the form of hydraulic fracturing with limited interactions to natural fractures (HF2) and shear reactivation with seismicity limited to streaks/asperities (HS4).

We have predominantly interpreted bound sequences arising from the finite (and time/volume dependent) extent of a propagating fracture restricting available slip area. While this interpretation matches both expectation and observation, we can still envision a scenario in which a finite fracture could produce an unbound sequence. For example, in the case that the finite extent of the fracture grows faster than the equivalent area of an unbound MLRG. In this sense, while the sequence is technically bound, it would never be empirically inferable by magnitude information. Criteria for this condition are given below: for the Vn generalization of MMAX as well as the tensile-like and shear-like fractures. There, the expected MLRG formulation is a combination of the population expectation (van der Elst et al., 2016) coupled with a proportionality between injected volume and event counts (Shapiro et al., 2010). Note that Σ is the seismogenic index, a parameter indicating a fault's potential to induce earthquakes per unit of injected fluid volume (Shapiro et al., 2010). 22n3-1blog⁡10V(t)+23log⁡10c-9.1≫Σb(Vngeneralization)31-1blog⁡10V(t)+23log⁡10167Δσsd3/2-9.1≫Σb(tensilefracture)423-1blog⁡10V(t)+23log⁡10167Δσk-9.1≫Σb(shearfracture)

As a reminder of terms: V(t) is the injection volume time-series, n is the volume exponent, b is the b-value, Δσ is the stress drop, while c, sd and k are injection propagation coefficients for different growth.

We have predominantly interpreted unbound sequences arising from fractures interacting with relatively-large pre-existing faults. While this matches our expectation and observations, we can still envision scenarios where pre-existing faults could produce bound sequences. For example, MLRG could be limited by the pre-existing faults being too small, the in situ stress resolved on the fault (Gischig, 2015; Norbeck and Horne, 2018), time-dependence of stress perturbations (Segall and Lu, 2015), fault geometry/complexity (Lee et al., 2024), fault roughness (Maurer et al., 2020; Wang et al., 2024), material/stress heterogeneity (Kroll and Cochran, 2021), rheology, or the extent of asperities. In fact, HS4 at the GTS is one such exception: where streaks outlined the rheological boundary between seismic asperities hosted within an aseismic fault. If these factors are limiting MLRG, CAP-tests should be able to discern their effects.