Introduction

The insatiable demand for energy continues to grow each year. In recent decades, hydraulic fracturing (HF) has been revolutionary for unlocking trapped hydrocarbon energy sources: fluids injected under high-pressure cause subsurface rocks to fail, stimulating fractures that act as permeable fluid pathways needed to extract resources1. This stimulation process can either propagate failures in tension or in shear; these failures can happen on either pre-existing fractures or newly created fractures. However, the urgency for more energy is also complicated by climate goals and decarbonisation needs. Enhanced Geothermal Systems (EGS)2 could play a part in the energy transition, since they have the potential to fill this niche by producing abundant, green, sustainable, baseload power3. EGS taps the limitless geothermal potential of the deep earth through stimulation techniques (e.g., HF)4: stimulated fractures can also act as fluid pathways needed for heat exchange. EGS could be on the cusp of widespread adoption, considering recent successes5. However, the deployment of HF techniques is not without risks.

The HF process also has the potential to induce earthquakes6,7. If stimulated fractures intersect a pre-existing fault, then the increase of pore pressure can cause fault slip reactivation8. These earthquakes can be consequential via economic/human losses, premature project termination, or resource development moratoriums9,10. For example, magnitude ~5 earthquakes have caused hundreds of millions of dollars in economic losses from both petroleum and EGS operations using HF10,11. On the other hand, there are also numerous HF projects that did not encounter detrimental earthquake impacts12,13,14,15. Thus, there is a clear need to effectively manage these earthquakes and mitigate their consequences.

Guidelines to managing induced earthquake risks exist16,17; these provide templates for pre-screening assessments, risk evaluation, monitoring, and defining operational endpoints. The core risk management framework is the traffic light protocol, which details how an operation should proceed, depending on the largest magnitude observed18. Risk-based approaches can identify magnitude thresholds for when an operation must stop, to remain below acceptable risk tolerances19,20,21. Despite these efforts, state-of-the-art procedures can still fail to robustly discern and then react to risky/consequential projects16,17 – as evidenced by high-profile cases that caused nuisance, damage, human losses, premature project termination, or development moratoriums9,10,11.

Fundamental to this risk management process is accurately anticipating the growth of the largest earthquakes22. Unfortunately, this is a poorly understood topic in seismology. What is generally accepted, is that the statistics of earthquake magnitudes (M) typically follows the Gutenberg-Richter magnitude frequency distribution (GR-MFD)23. A natural consequence of the GR-MFD is that the largest magnitude events (MLRG) scale with the size of the catalogue (N)24. It is debatable whether any other physical process bounds the growth of magnitudes: i.e., an upper bound magnitude (MMAX) that restricts the (statistically expected) growth of magnitudes. Some MMAX models have been hypothesised. In the simplest model, an MMAX can arise from the finite size of the fault extent, since magnitude is tied to shear slip across a fault area25. More contentiously, MMAX could also be some injection-dependent process that changes with time; for example, either evolving with seismic moment release (McGarr-like models)26,27 or scaling with self-arrested rupture dynamics (Galis model)28. Investigating the physics of these MMAX models would benefit from approaches that could verify/falsify their presence in a catalogue.

However, there is a lack of empirical approaches to identify/falsify these MMAX models – the identification of MMAX solely from a catalogue is fraught with difficulty. Traditional approaches to this MMAX problem have found it to be solvable only in ideal conditions29,30, often requiring a significant degree of truncation for robust identification. On the other hand, verifying the existence of MMAX within an induced earthquake sequence is significant, because then the risk of induced seismicity is inherently more controllable22. Given this gap between need/difficulty, expert opinion panels on MMAX31 often substitute for quantitative science out of necessity.

In this study, we consider a reformulation of this MMAX problem. Here, we implement a novel suite of statistical tests to discern if a sequence of earthquake magnitudes was bound by an MMAX32. These tests compare event magnitudes (M) against ‘jumps’ in the sequence of largest earthquakes (ΔMLRG), to infer the presence of a maximum upper bound (MMAX). Simply put, the statistical distributions of M and ΔMLRG are identical when a sequence is unbound, but distinguishable in the presence of an MMAX (Fig. 1)—this is because the presence of an MMAX restricts the allowable sizes of ΔMLRG. Our suite of three statistical tests exploits this fact. The first test uses the Kolmogorov-Smirnov test (KS-test) to compare M and ΔMLRG observations, to discern if they are drawn from the same distribution; this tests for the existence of an MMAX. The second test uses a maximum likelihood estimation (MLE-test) to quantify the value of MMAX. The third test computes ensemble weights (EW-test) through Akaike and Bayesian Information Criteria; this selects the best MMAX model amongst several hypothesised choices, given the catalogue data. We refer to this suite of three novel tests as CAP-tests. These CAP-tests are generally more sensitive to MMAX bounds than traditional approaches32. Full details on terminology, methods, guidelines to the interpretation of results, and synthetic examples are all available in the Supplementary Information (Text S1–S4).

Fig. 1: Schematic demonstration of observational and methodological concepts.
figure 1

The upper panels show hypothetical catalogues of earthquakes. Earthquake magnitudes (M, pink circles), the sequence of largest events (MLRG, red line), and jumps in the largest event magnitudes (ΔMLRG) can be directly observed. Potentially, some unobservable physical process could be bounding these catalogues (MMAX, blue line). Left and right panels detail contrasting concepts for an unbound and bound catalogue, respectively. If there is an MMAX, then it can be statistically inferred from the observables. Middle panels show the differences between distributions of M and MLRG, when bounded by MMAX (or not). The distribution of M follows a GR-MFD: analytical cumulative (thick black line) and non-cumulative (thin black line) agree with numerical cumulative (pink line) and non-cumulative (pink bars) distributions. The distribution of ΔMLRG will differ from the GR-MFD when bounded by MMAX. Bottom panels show the results of EW-tests using these concepts. Weights of two hypothesised models, an unbound MMAX (pink area) and a bound MMAX (blue area), change as new values of MLRG are observed. EW-tests can quickly infer the presence of the true MMAX model, from the equivocal a priori assumption (white horizontal line). Evidence for a model ‘winning’ over another becomes substantial/positive, strong, and decisive when the ratio-of-weights (i.e., odds ratio) is 3, 10, and 100, respectively.

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We demonstrate how this workflow (Fig. 2 & S1) can be used to identify concerning cases of induced seismicity. This is accomplished by contrasting cases of induced seismicity related to HF (Fig. S7). In the first case, we consider Preston New Road 1z (PNR-1z)33: a shale gas HF operation in the UK that injected ~4200 m3 over 17 (of 41 planned) stages and produced earthquakes up to ML 1.6 in 2018. In the second case, we consider Helsinki St115: an EGS operation in Finland that injected ~18,200 m3 over five stages and produced earthquakes up to MW 1.9 in 2018. Our tests find that all stage stimulations in these two cases follow a bound process, dominantly suggesting fracture growth via stage stimulations. In the contrasting cases, we consider data from the Frontier Observatory for Research in Geothermal Energy (FORGE)34 and Preston New Road 2 (PNR-2)35. FORGE is an EGS operation in Utah that injected ~18,500 m3 over ten stages and produced earthquakes up to MW 1.8 in 2022 and 2024. PNR-2 is a shale gas HF operation in the UK that injected ~2500 m3 over seven stages and produced earthquakes up to ML 2.9 in 2019. Our tests find that the first handful of stages follow a bound process (i.e., restricted by a volume-based MMAX). However, the subsequent (and problematic) stages were clearly unbound, dominantly suggesting fault reactivation via stage stimulations. The PNR-2 case was prematurely terminated because of induced seismicity concerns.

Fig. 2: Logical sequence of statistical tests to discern MMAX bounds.
figure 2

Simple-tests are used to pre-screen for potentially bound catalogues by considering the degree-of-truncation. Afterwards, CAP-tests can more rigorously identify bound cases.

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Bound growth

Bound growth at PNR-1z

HF was used in the UK at three wells, to extract the hydrocarbons locked within the Mississippian aged Bowland Shale. The operation at PNR-1z, near Blackpool, was the second HF operation in the UK33. The first set of stimulations (October–November 2018) produced moderate seismic response; these events were concerning due to their magnitude above the red-light threshold, which caused an operational pause. The second set of stimulations (December 2018) resumed after this pause. The largest event (ML 1.6) occurred on 11 December 201833. All events appear to be hydraulically linked into a single cluster (Fig. S8; Text S5.1). The success of operations at PNR-1z enabled the subsequent operation at PNR-2.

To begin assessing if some process might be restricting magnitude growth, we fit the GR-MFD23 and perform the simple-tests (Text S5.1). The earthquakes here appear to be deficient in large magnitude events (Fig. S9). Correspondingly, the discrepancy between the largest observed event and the largest expected event magnitudes is −0.5 MW, which is a rare occurrence ( ~ 2%) purely by chance. These initial assessments are suggestive of some MMAX upper bound restricting the PNR-1z catalogue growth.

Next, we use the CAP-tests to detect and assess the potential for MMAX more rigorously (Fig. 3 & S10). Average KS-test confidences are 99.82% (Text S5.1), where 95% is a common threshold to declare statistical significance. Next, we seek test hypotheses for the functional form of MMAX: our EW-tests focus on a McGarr-like model (i.e., log-proportional to injected volume; \({{{\rm{M}}}}_{{{\rm{MAX}}}}\propto {\log }_{10}\left({V}^{1}\right)\))26,27, the Galis-like model (i.e., \({{{\rm{M}}}}_{{{\rm{MAX}}}}\propto {\log }_{10}\left({V}^{3/2}\right)\))28, and the unbound null hypothesis. Similarly, EW-tests also show strong evidence for an MMAX bound process. By ~100 event observations, respective model weights are already {0.39, 95.69, 3.92%} which means the Galis-like MMAX model is ~24 times more likely than unbound (log10 odds ratio of +1.4; Fig. 3a). By the end of stimulations, the Galis-like model is >100 times more likely than the unbound model (Fig. 3a)—which surpasses the definitive evidence threshold.

Fig. 3: EW-tests applied to PNR-1z.
figure 3

a EW-tests are shown for all the stages at PNR-1z. The input data includes earthquake magnitudes (circles), the sequence of observed MLRG (red line), alongside the expected MLRG at the 10/50/90 percentiles (red area); two possible injection-based MMAX models are considered (blue lines). Correspondingly, the ensemble weights from the EW-test are shown; bound models are highly favourable. b Map view showing the locations of earthquakes (circles) alongside the well trajectory (black line) and stage locations (grey lines). Throughout the figure, all earthquakes are colour-coded accordingly to their likelihood of being bound or unbound from EW-tests.

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Bound growth at Helsinki St1

HF was used in a deep geothermal well to create an EGS in the Precambrian Svecofennian basement rocks near Helsinki, Finland. The St1 operation injected ~18,200 m3 of fluid within five stages from June-July 201815. These stimulations produced a moderate seismic response; the largest event (MW 1.9) occurred on 8 July 2018. All events appear to be hydraulically linked into a single cluster (Fig. S11; Text S5.2), which is consistent with prior interpretations15. The success of this operation allowed for later restimulation of the reservoir.

To assess if some process might be restricting magnitude growth, we first employ the simple tests based on GR-MFD analyses (Text S5.2). The earthquakes here appear to be deficient in large magnitude events (Fig. S12). Correspondingly, the magnitude discrepancy between the largest observed event and the largest expected event is -0.7 MW, which is a rare occurrence ( < 1%) purely by chance. These initial assessments are suggestive of some MMAX upper bound restricting the St1 catalogue growth.

Next, we use the CAP-tests to detect and assess the potential for MMAX more rigorously (Fig. 4 & S13). Average KS-test confidences are 98.17% (Text S5.2). Similarly, EW-tests also show strong evidence for an MMAX bound process (Fig. 4): McGarr-like and Galis-like models are >100 (log10 odds ratio of +2.2) and ~57 (log10 odds ratio of +1.8) times more likely than unbound models, respectively. This dataset is somewhat ambiguous between either McGarr-like or Galis-like models best explaining the data.

Fig. 4: EW-tests applied to Helsinki St1.
figure 4

a EW-tests are shown for all the stages at Helsinki St1. The input data includes earthquake magnitudes (circles), the sequence of observed MLRG (red line), alongside the expected MLRG at the 10/50/90 percentiles (red area); two possible injection-based MMAX models are considered (blue lines). Correspondingly, the ensemble weights from the EW-test are shown; bound models are highly favourable. b Map view showing the locations of earthquakes (circles) alongside the well trajectory (black line) and stage locations (grey lines). Throughout the figure, all earthquakes are colour-coded accordingly to their likelihood of being bound or unbound from EW-tests.

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We note that we have examined supporting cases, including the EGS stimulations at Soultz-sous-Forêts in France36. There, we also find some evidence for a bound process restricting the growth of earthquake magnitudes during their stimulations (Text S5.5; Figs. S27–S36).

The transition to unbound growth

The transition to unbound growth at FORGE

Utah FORGE is a geothermal experiment ( ~ 16 km NE of Milford) aimed at commercialising EGS34. Two deviated wells form a doublet heat exchange system (16 A injector & 16B producer). The first three stages of well 16 A were stimulated in 2022; in 2024, stage 3 was restimulated, followed by the remaining stimulations (Figs. S14-S15). Originally, nine stages were planned for well 16 A at FORGE. However, frack barriers were encountered at stage 6, preventing fracture stimulation; in response, a tenth stage was added. Correspondingly, stage 6 separates the adjacent stages into two hydraulically separate compartments. Overall, stages are compartmentalised into hydraulically independent clusters as follows: cluster 1 (stages 1-2), cluster 2 (stages 3–6), and cluster 3 (stages 7–10). Additional justifications for this clustering choice are provided in the Supplementary Information (Text S5.3) and in prior studies35.

To assess the potential for bound/unbound earthquake growth, we first employ the simple GR-MFD analyses (Text S5.3). The earthquakes here appear systematically skewed in larger magnitude events: both in terms of the full catalogue (Fig. S16) and on a cluster-by-cluster basis (Fig. S17). Furthermore, we also consider the discrepancy between the largest observed event and largest expected event magnitudes. Correspondingly, this difference ranges between -0.4 and -1.3 MW for the first two clusters, but is +0.1 MW for the third cluster. These are rare occurrences ( ~ 1%) for the first two clusters. Together, these initial assessments are suggestive of some MMAX upper bound restricting the FORGE catalogue growth for clusters 1-2.

To more rigorously examine MMAX upper bounds, we use the suite of CAP-tests. Both KS-test and MLE-test results provide strong evidence for the existence of MMAX upper bounds. Specifically, KS-test confidences range between 99 and 100% for clusters 1 and 2. Thus, there is strong evidence for some physical MMAX process. On the other hand, cluster 3 does not meet the 95% confidence threshold. For the EW-tests on stage 3 in 2022, within ~100 event observations, respective model weights are already {70.44, 17.69, 11.87%} which means that the McGarr-like MMAX model is ~6 times more likely (log10 odds ratio of +0.8; Fig. 5a). Furthermore, clusters 1 and 2 show strong evidence for a bound process after EW-tests (Fig. 5). Specifically, the McGarr-like model appears to best explain these bound clusters (Figs. S18–S21). On the other hand, the EW-tests for cluster 3 show strong evidence for an unbound process: the unbound model is ~50 times more likely than the best-fitting bound model. These results hold, even after various perturbation/sensitivity tests (Text S6–S7).

Fig. 5: EW-tests applied to FORGE.
figure 5

EW-tests are shown for a the 2022 stage 3 and b for the 2024 cluster 3 (stages 7–10). The input data includes earthquake magnitudes (circles), the sequence of observed MLRG (red line), alongside the expected MLRG at the 10/50/90 percentiles (red area); two possible injection-based MMAX models are considered (blue lines). Correspondingly, the ensemble weights from the EW-test are shown; bound models are highly favourable in most cases (Figs. S18–S21). c Map view showing the locations of earthquakes (circles) alongside the 16 A well trajectory (black line) and stage locations (grey lines). Throughout the figure, all earthquakes are colour-coded accordingly to their likelihood of being bound or unbound from EW-tests.

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The transition to unbound growth at PNR-2

The operation at PNR-2, near Blackpool, was the most seismically problematic37 and last of the three HF operations in the UK. The first 1–3 stages only produced moderate seismic response in the western-most clusters, while more concerning seismicity was encountered at the eastern-most clusters, which predominantly occurred during stages 5–7 (Fig. S22). The eastern-most cluster culminated in the ML 2.9 event on 26 August 2019, which occurred >72 hours after the shut-in of stage 737. This event caused the premature termination of PNR-2, after only the first seven stages (of 47 planned) were completed. Ultimately, these earthquakes spurred a resource development moratorium in the UK38.

To assess the potential for bound/unbound earthquake growth, we repeat the simple GR-MFD analyses. The western-most clusters appear to be deficient in large magnitude events, while the eastern-most cluster is abundant (Figs. S23-S24). Correspondingly, the discrepancy between the largest observed event and largest expected event magnitudes is -1.2 MW and +0.8 MW, respectively. This is a rare occurrence ( < 1%) for the western-most cluster, but not the eastern-most. Together, these initial assessments are suggestive of some MMAX upper bound restricting the PNR-2 catalogue growth at the western-most cluster, but not the eastern-most.

Again, we use the CAP-tests to detect and assess the potential for MMAX more rigorously (Fig. 6). KS-test and MLE-test results show strong evidence for MMAX in the western-most case (i.e., > 99.99% confidence), but not the eastern-most case (Text S5.4). Similarly, EW-tests also show strong evidence for a bound process, but only for the western-most cluster (Fig. 6a). The western-most cluster is somewhat ambiguous between either McGarr-like or Galis-like models best explaining the data. On the other hand, the eastern-most cluster definitively indicates an unbound process, being ~100 times more likely than the best-fitting bound model.

Fig. 6: EW-tests applied to PNR-2.
figure 6

EW-tests are shown for a the western-most and b eastern-most clusters. The input data includes earthquake magnitudes (circles), the sequence of observed MLRG (red line), alongside the expected MLRG at the 10/50/90 percentiles (red area); two possible injection-based MMAX models are considered (blue lines). Correspondingly, the ensemble weights from the EW-test are shown; the seismic response is spatially partitioned into eastern/western clusters, where stages sequentially transition from bound into unbound (Figs. S25–S26). c Map view showing the locations of earthquakes (circles) alongside the PNR-2 well trajectory (black line) and stage locations (grey lines). Throughout the figure, earthquakes are colour coordinated accordingly to their likelihood of being bound or unbound from EW-tests.

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Implications

Practical risk management

Induced earthquakes pose considerable risks. However, the chances of encountering earthquake consequences for a bound/unbound stage stimulation are significantly different (Text S8; Fig. S41). Thus, the ability of CAP-tests to discern between these two extremes constitutes an important reactive risk mitigation measure: bound stages could proceed as planned, while unbound stages could be treated with caution (or even prematurely stopped and then skipped). This would be especially important for cases where earthquake magnitude growth is approaching the red-light19,20,21—the last-possible stopping-point before a regulatory intervention is required. In this sense, CAP-tests provide information on how/if ‘jumps’ in earthquake magnitudes (or trailing seismicity) are restricted, which is consistently identified as an important factor for managing induced seismicity22,39,40.

To elaborate, we separate bound/unbound clusters using just the EW-test results (Fig. S42). We use an odds ratio threshold of 3-to-1 (i.e., log10 of +0.5) to discern a bound case to statistical significance; all subsequent ΔMLRG events are labelled bound. Clusters below this threshold are considered unbound. We then examine the distribution of ΔMLRG from these two catalogue subsets (Fig. 7); we find that they are different to statistical significance (KS-test confidence 99.8%). The unbound cases are well-fit to the GR-MFD, with a typical b-value of 1.73 ± 0.18 (R2 0.989) and ΔMLRG magnitude jumps up to +1.2. On the other hand, the bound cases are not well-fit to the GR-MFD, with an ‘unphysical’ b-value of 3.02 ± 0.30 (R2 0.967) and ΔMLRG abruptly truncated near +0.3.

Fig. 7: Comparing ΔMLRG distributions.
figure 7

Clusters were separated into bound (blue) or unbound (pink) groups using EW-tests. The ΔMLRG data are plotted as both the cumulative (circles) and non-cumulative (bars) distributions, alongside the GR-MFD fit to the data (solid line). A hypothetical traffic light protocol (dashed line) informs a confusion matrix (inset), that delineates the amount/proportion of bound/unbound events that might trigger the red-light.

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First, this finding is noteworthy, since it provides positive feedback that the CAP-tests are working correctly. Theoretically, unbound ΔMLRG values are expected to follow the same distribution as their underlying magnitudes; on the other hand, bound ΔMLRG values deviate from their GR-MFD32—because MMAX restricts bound ΔMLRG values, creating apparently steeper b-values (Fig. 1). This expected difference in ΔMLRG distributions is consistent with the observations from our EW-test separation (Fig. 7). More generally, this biasing phenomenon could potentially explain the anomalously high b-values typically associated with earthquakes encountered during HF stimulation41, since a temporally growing MMAX produces a long and pronounced roll-off at the high-magnitude tail of the GR-MFD.

Furthermore, this finding is significant, since these bound cases are appreciably less hazardous. For example, we can consider a hypothetical situation where the operator approaches a red-light. If MLRG was currently 0.30 M away from the red-light, the next largest event would trigger the red-light in ~30% of unbound cases but only ~4% of the bound cases (Fig. 7). This is especially important considering that earthquake consequences scale exponentially with magnitude42. Overall, this hypothetical scenario demonstrates how CAP-tests can be used to quantify magnitude jumps, providing insights into red-light designs and operational decision-making. Correspondingly, we note that operations that encountered significant consequences or were prematurely terminated (e.g., Basel, Pohang, PNR-2) tended to encounter unbound earthquakes32.

This approach could find near-future use in post hoc analyses between stage stimulations – especially in tandem with completion engineering information. Potentially even providing complementary information for identifying hydraulic connectivity between stages. While we have used CAP-tests to demonstrate this approach on HF and EGS cases, we anticipate this approach will be useful for risk management of other types of induced seismicity (e.g., disposal, mining, reservoir impoundment) as well as tectonic earthquakes. Furthermore, McGarr/Galis predictions of small magnitude MMAX at low volumes suggest the potential to test these models at smaller scales, or use deep catalogues to improve warning times. With further development, this type of approach could be used to quantify logic-tree weights of MMAX within seismic hazard analysis31 or to inform seismicity forecasts with real-time data. Although, we caution that the potential for a bound-to-unbound transition (Fig. 8) needs to be considered. We expect CAP-tests to be an important part of future tools that de-risk induced seismicity.

Fig. 8: Interactions between stimulated fractures and fault reactivation during hydraulic fracturing.
figure 8

A series of HF stages of increasing complexity (text labels) are considered alongside a hypothetical well (black & grey rectangle). In the simple case, the host rock is initially split in tension via stimulated fractures (grey polygons) and may also subsequently reactivate in shear. In the complex case, small pre-existing faults (black polygons) can also be reactivated in shear slip. The clustered case hydraulically connects a series of stages/perforations into a single fracture/fault network. In the last case, stimulated fractures intersect a large fault system that is critically-stressed for slip. By watching the growth of earthquake magnitudes, CAP-tests can discern between bound cases (blue circles) and unbound cases (pink circles).

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Stimulation optimisation

Distinguishing between bound fracture growth and unbound fault reactivation is also significant from the perspective of reservoir optimisation. Stage fluids diverted into faults hinder stimulation efficiency and thus limit reservoir productivity43. The use of CAP-tests could be paired with complimentary diagnostic tests attempting to discern the presence of faults44. Future operators could use these diagnostic tests to proactively discern stages that are linked to faults, thus avoiding stimulation into unproductive and seismically risky stages. On the flip side, operators that recognise a bound case could potentially continue injecting closer to the red-light – allowing for greater fracture stimulation, and greater reservoir productivity, which enables further energy development. For example, the next largest event for ~50% of bound cases would trigger a red-light if MLRG was 0.10 M away from the red-light (Fig. 7). Although, this would require further research and validation testing into the nuances and efficacy of CAP-tests (e.g., Text S6-S7). A hypothetical scenario could be a bound-to-unbound transition; for example, if bound fracture growth suddenly encounters a large fault (Fig. 8). Generally, having a strong understanding of the fault architecture and hydraulic connectivity appears to be important for discerning bound cases.

Physical mechanisms

The operation at PNR-1z, St1, and initial stages of FORGE/PNR-2 appear to have responded in similar manner to stage stimulation; all have encountered some physical process bounding magnitude growth. There is strong evidence for this from the GR-MFD and CAP-tests, that cannot be reasonably explained by statistical variance (Sections 2–3; Figs. 3–7). This means that there is a measurable impact of finite fault size restricting magnitudes in these cases, albeit with a time dependence from the injection.

Certainly, the spatial extent of stimulation earthquakes increases with time during HF operations, including FORGE45,46. This finite extent of the stimulated fractures provides a potential candidate mechanism for restricted magnitude growth. As well, there is growing evidence suggesting that injection-based processes have restricted at least some stage stimulations at HF and EGS operations16,47,48,49 and during laboratory experiments50. Correspondingly, our results indicate that either Galis-like or McGarr-like models26,27,28 can explain the data, depending on the case. This result also corresponds with the scaling of total seismic moment and total injected volume observed during HF operations in Canada47. We note that while Galis-like and McGarr-like are the ‘winning’ models, there could still be other unknown/untested models that explain the data even better. Future high-resolution studies (aided by CAP-tests) will be able to further elucidate the physics of the bounded fracturing process.

On the other hand, cluster 3 at FORGE and the eastern-most cluster at PNR-2 have responded in a fundamentally different way; there appears to be no (currently discernible) bounds restricting the growth of those earthquake magnitudes. Correspondingly, other studies of PNR-2 have also recognised sequential changes in stage stimulations towards the eastern-most cluster21,51. We argue that this likely represents fault reactivation: the fluid/pressure diffusion process triggers slip initiation on a large fault, releasing the tectonic stresses stored over geological timeframes (Fig. 8). The earthquake magnitudes were able to grow unrestricted by any physical limitations. This interpretation is consistent with numerous other large magnitude cases that have also exhibited unbound growth, which appears to be the norm33. Further work would be needed to better understand the processes responsible (e.g., fracture complexity, hydraulic connectivity, stress orientations) for creating this bound-to-unbound transition52.

In this sense, we argue that CAP-tests represent an approach able to empirically discern between triggered or driven induced earthquakes22,53,54. Earthquakes induced during bound fracture growth are measurably discernible from those triggered during fault reactivation. This approach will be useful in testing against other cases that appear to be deficient in high magnitude events55,56.

Future directions for CAP-tests

We note that our CAP-tests are limited, in the sense that they can only infer the presence of MMAX and then select the best model among the ensemble of MMAX proposals. If the true model is missing from this ensemble, then it is impossible to select. Here, we discuss the potential for this oversight bias in our study, as well as other errors.

To examine this, we focus on the stage 3 stimulation at FORGE (Fig. 4a; cluster 2a). We use the previously fitted McGarr-like and Galis-like MMAX models to build the expected GR-MFD at the end of catalogue recording. To do so, we use a mixture model that performs a summation of truncated GR-MFDs, with the MMAX temporally varying over the course of injection. This allows us to visualise the expected roll-off that the McGarr-like and Galis-like MMAX ought to have imposed on the data (Fig. 9). The gap between data-and-expectation at high magnitudes provides a qualitative sense of model misfit – further indicating as to why the EW-tests preferred McGarr-like to Galis-like to unbound, respectively.

Fig. 9: Comparison of expected fits to the GR-MFD.
figure 9

Data are plotted as both the cumulative (circles) and non-cumulative (bars) distribution, alongside the magnitude-of-completeness Mc (dashed lines). Expected fits to the data based on the unbound (pink line), Galis-like (pale blue dashed line), McGarr-like with a truncated GR-MFD (dark blue solid line), and McGarr-like with a tapered GR-MFD (dark blue dashed line). Data is from stage 3 stimulation at FORGE in 2022 (cluster 2a). The magnitude-of-completeness Mc is also shown (dashed lines).

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Despite these efforts, there still remains a misfit gap. On one hand, this gap could be the result of an oversight-bias, which would suggest that a more conservative MMAX-volume relationship is still needed. On the other hand, additional physical complexities like limited stimulated rock volume, a distribution of fault/fracture areas, or seismic moment conservation could have contributed to this gap. Our examination here considered truncated GR-MFDs, if we instead consider a mixture of tapered distributions57, then this gap appreciably closes. We note that tapered GR-MFDs have been introduced as a potential explanation for the aforementioned physical considerations, and are generally viewed as advantageous over truncated distributions57. Regardless, CAP-test can accurately infer the presence of this roll-off. Further interpretation between these physical mechanisms will require additional studies and information to make more sophisticated MMAX proposals.

Last, there is the potential for errors lurking within the catalogue. These can be related to differing measurement types, differing catalogue building, or magnitude uncertainties, all of which can appreciably impact interpretation58. In light of these errors, we have tried to be careful in our analysis, to consider magnitude dithering, bootstrapping and parameter perturbations (Text S6–S7). We note that the relative importance of ΔMLRG for discerning MMAX provides future directions for a quality-control prioritisation scheme. In cases where constraining MMAX via CAP-tests are important, quality-control efforts should primarily focus on accurately capturing the MLRG envelope (Fig. 1 & S1).

Conclusions

We find that our CAP tests can confidently discern cases of bound earthquake magnitude growth during HF stage stimulations at PNR-2 and FORGE (including PNR-1z, ST1, and Soultz-sous-Forêts). This includes an identification of a transition into (problematic) unbound growth, which prematurely terminated operations at PNR-2. We discuss the potential physical mechanisms controlling this binding process. We also discuss the implications for operational risks regarding reservoir optimisation and induced seismicity, including our thoughts on possible mitigation measures guided by CAP-tests. These tests could be used to de-risk future operations, aiding energy development and the transition to a green energy economy.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.