Introduction

The frequency and detection of induced earthquakes has increased dramatically over the past few decades, leading the scientific community and industry to become more aware of the issue. Most sources of induced seismicity involve the underground injection or withdrawal of fluid1. Earthquakes have been associated with several human activities ranging from oil and gas extraction2,3,4 to underground wastewater storage5,6, and even artificial lakes for hydro-power7, tunnelling8,9 and nuclear waste disposal10.

Among the various industrial operations, geothermal exploitation constitutes a promising approach for green energy production, but often requires fluid stimulation to increase the reservoir’s permeability to create Enhanced Geothermal Systems (EGS11,12). Such stimulation activities, however, are often linked to strong seismic events. Numerous cases around the world have led to much debate in the scientific community and industry as to whether this technology is really beneficial. Renowned examples are: M 2.9 in Soultz-sous-Forêts (2003, France)13, ML 3.4 in Basel (2006, Switzerland)14 and Mw 5.5 in Pohang (2017, Republic of Korea)15; but other geothermal exploitation schemes are not exempt from damaging induced earthquakes, as a recent sequence in Strasbourg culminating in an MLv 3.6 shows (2019–2021, France)16.

The occurrence of induced earthquakes of potentially large magnitude requires the development of strategies and tools to mitigate the hazard posed by EGS stimulations17,18. Numerical models constitute the basis for the most advanced tools proposed to understand, forecast, and mitigate the hazard and risk accompanying geothermal stimulations. Multiple classes of models have been used to investigate fluid induced seismicity19, from the purely statistical20,21,22 to the fully coupled thermo-hydro-mechanical23,24,25. Hybrid models strive to combine the strength of both approaches, using the velocity of statistical methods, and the complexity of the principal physical processes26,27,28,29,30,31. A fundamental aspect of induced seismicity mitigation relies on linking modelling results and risk/hazard calculations32,33,34.

Numerical models can also investigate the impact of injection scenarios on the risk posed by induced seismicity31,35,36,37. In this study we use the capabilities of a loosely coupled hydro-mechanical-stochastic model – TOUGH2-Seed38 – to investigate how seismicity is affected by the presence of a major fault zone and to understand if a different injection strategy may help with predicting the potential hazard. To assess an injection scenario, we follow a simple metric: the evolution of the Gutenberg–Richter b-value39, describing the relative size distribution of earthquakes, during both injection and post-shut-in phases. This approach allows us to go beyond statistical models that aim to simulate the rate/number of events (λ or the Gutenberg-Richter a-value)32,40,41 by modelling both the a-value and b-value. The approach is based on the assumption that the b-value is a proxy for the state of stress in the subsurface: indeed, several studies have shown that the b-value is sensitive to the state of stress (inverse dependence observed both in laboratory42,43,44 and field45,46 studies).

Modelling both the number of events and the b-value provides an additional proxy for seismic risk47,48. Indeed, the b-value is the true driver behind the level of risk/hazard as highlighted in Fig. 1a: for a comparable a-value, Pohang and Basel have vastly different probabilities of a magnitude 6 occurring. Usually, the probability of occurrence of a large event is used with a fixed b-value calculated for a region, as the b-value shows variability depending on the tectonic context (Fig. 1a). However, more recent studies with high-resolution catalogues, both in natural contexts49,50 and in induced seismicity contexts5,51, have shown that the b-value changes through time and space within a given region due to additional forcing (e.g. other earthquakes or fluid injection).

Fig. 1: The b-value varies from site to site, and in time.
figure 1

a Frequency-Magnitude Distribution of selected EGS and scientific injection activities. The thick line stops at the maximum observed magnitude, while the dashed line shows the extrapolation to higher magnitudes14,59,63,69,80,81,82. All magnitudes are converted to Mw, see Supplementary Table S1 for details. b Temporal evolution of the b-value with a high-resolution catalogue in Basel during the December 2006 geothermal stimulation. Time zero corresponds to the start of the full injection phase on 2 December 2006 at 18:00. The events are binned in 0.2 days windows with a 0.1 days overlap. The shaded area show the ± one and two standard deviation of the calculated b-value.

Full size image

These changes in the b-value (both spatial and temporal evolution) have been shown to help with the forecasting of larger events: the b-value tends to increase after a main shock and during the aftershock sequence49. However, it has been observed that in the case of doublets, the second event can be preceded by a decrease in the b-value (compared with the background) and a Traffic Light System based on b-value variation to understand when the hazard has passed has been suggested50. Thus, the b-value can be used as a proxy for average stress conditions and as a tool for risk mitigation to discriminate between a typical aftershock sequence and a precursory sequence. In the case of induced seismicity in Basel, initial studies showed how the b-value dropped significantly after shut-in21, but with a high-resolution catalogue51, the b-value is actually observed to drop days before the shut-in and the two largest events (ML 2.9 during injection and ML 3.4 a few hours after shut-in14). Figure 1-b shows the evolution of the b-value in time in Basel using a high-resolution catalogue51 with a fixed time windows of 0.2 days. Supplementary Fig. S1 presents a more complete analysis of evolution of the b-value, number of events above completeness and magnitude of completeness in time for the Basel case. The variability in the b-value during the injection and a sharp drop around day 3 have been thought to be linked to pore-pressure perturbations and a change in the growth behaviour of the seismic cloud. These changes in the b-value during and after the active injection phase suggests that the injection activity can somehow influence the state of stress and the b-value as a consequence.

In light of this observation, in this study we attempt to model a simplified Basel-like injection (with injection rates increasing in steps) to understand the physical mechanisms driving the changes in the b-value. We then test different injection strategies in an ideal EGS setting, delineating which scenario is more prone to show a potential hazard in the early stages of injection, in order to provide relevant insights for operators.

Results and discussion

In order to reproduce a typical EGS system and simulate an earthquake catalogue resulting from stimulation, a TOUGH2-Seed model is defined as a simple 4-km-deep injection placed in a 1.5 × 1.5 km reservoir with 2 km of thickness centred around the injection. The potential hypocentres (aka seeds) are randomly distributed within this physical volume, a set of stochastic realisations being used each time to account for this randomness aspect. Details of the model implementation can be found in the Methods section.

A base case simulation expands on previous modelling of a Basel-like setting27, with the addition of pressure-dependent permeability changes and an extension of the geomechanical model with a full 3D stress tensor formulation. We use a linear relationship between b-value and differential stress52, adapted to fit the amplitude of the b-value observed in Basel (Fig. 1b51; see the Methods section for more details). Cold water is injected for 5 days with the rate increasing from 10 kg⋅s−1 to 60 kgs−1. The step-increasing injection pattern is chosen to represent a classical approach to injections for EGS with rates similar to those injected in Basel14 (Fig. 1b). The shut-in occurs after 5 full days of injection, and we model the evolution up to a total of 10 days.

On top of this homogeneous base case, we also look into a reservoir containing a major fault zone close to the injection, similar to the fault observed in Basel, where seismicity started aligning a few days into stimulation53. Indeed, fault zones represent high-risk features capable of fostering large events and often go undetected by initial assessments of the site. A fault zone is simulated by changing both hydraulic (i.e. permeability) and geomechanical properties (i.e. density of seeds, b-value). More details can be found in the Methods section.

Hydraulic and seismological response to a step-increasing stimulation

Figure 2 presents a summary of the output of the base case model, with the evolution of the pressure and permeability changes at different monitoring points (solid lines in panels a and b respectively) and a map view of the pressure changes and the density of seismicity averaged over 48 realisations (panels c and d; Supplementary Fig. S2a shows the density of seismicity with depth). The evolution of the pressure at different monitoring points (Fig. 2a) shows an initial sharp pressurisation close to the injection and further gradual pressurisations happening over time with the propagation of the pressure front and the surge in pressure associated with increases in the injection rate. The permeability follows a similar pattern with an initial enhancement followed by successive increases after the injection rate increases, with each monitoring point further from the injection seeing a similar pattern, albeit with a delay (Fig. 2b).

Fig. 2: Modelling results for the step increasing injection.
figure 2

Changes in pressure (a) and permeability (b) during the injection cycle, dashed lines correspond to the case with a fault. Distribution of the pressure changes after 10 days for the cases without a fault (c) and with a fault (e). Density of the seismicity averaged over all realisations without a fault (d) and with a fault (f).

Full size image

In terms of the spatio-temporal distribution of the seismicity, a time-distance from injection plot presented in Supplementary Fig. S3a allows us to see previously observed and modelled characteristics, with a clustering of higher b-value events close to injection indicative of events occurring in previously reactivated/stimulated areas27, and the classical triggering and back fronts of seismicity40 indicative of pore pressure propagation in the system.

Furthermore, Fig. 2d gives a stacked and normalised view of the distribution of the seismicity in a 2D horizontal slice centred on the injection. The central region shows a dense seismic cloud, which gradually fades further away from the injection. The comparison of the pressure perturbation extent (Fig. 2c) and the average seismicity cloud (Fig. 2d) show that the seismicity is limited to the confines of the relatively highly pressurised zones.

The dashed lines in panels a and b of Fig. 2 present the evolution of the pressure and permeability changes for the fault case, while panels e and f show a map of the pressure changes and of the density of seismicity averaged over all realisations, respectively. The evolution of the pressure over time remains identical except for the monitoring point located 100 m from injection, as this monitoring point is located at the fault. The hydraulic fault accommodates more pressure and changes the distribution of pressure (Fig. 2e). The distribution of the seismic cloud follows the distribution of the pressure perturbation and shows that both the areas close to the injection and the central part of the faults exhibit high densities of seismicity. The time-distance plot in Supplementary Fig. S3b shows a dual behaviour of the seismicity and highlights the different hydraulic diffusivity of the fault with clearly distinct triggering and back-fronts of seismicity. The evolution of the permeability over time is mostly unaffected by the presence of the fault with the exception of the region in its immediate vicinity (Fig. 2b dashed lines). The average number of events for this fault case is similar to the previous case (referred to as the homogeneous case; see Table 1).

Table 1 Statistics on the different injection strategies without and with a fault.
Full size table

Figure 3 presents the evolution of the b-value and the hourly a-value (as described in the Methods section) in the top two panels for the homogeneous and fault cases. Both variables see an increase following each change in the injection rate: a systematic pulse-like variation for the b-value and an increase followed by a stable plateau for the a-value. The variation in the a-value is simply due to the change in the pressurisation rate in the system: the higher the flow rate, the higher the pressure, and so the stronger the pressure changes at the propagating front, resulting in a larger number of seeds reactivated. The variation in the b-value is associated with the intrinsic relationship between the b-value and the differential stress: as a seed is reactivated, the differential stress decreases (due to stress drop) and as a consequence the b-value increases. The peaks of the b-value are then associated with previously reactivated seeds (i.e. events occurring in already stimulated regions). Such repeating events are simply caused by the change in flow rate and pressure, causing the already activated seeds to reactivate again at a new, higher pressure. Figure 3c shows the temporal evolution of the average distance from the injection of the reactivated seeds (labelled ’average distance’), clearly showing a dip after each change in the injection rate, thus showing that seeds closer to the injection get reactivated and drive the b-value up following a change in the injection. The vertical distance (labelled ’average depth’) does not change substantially, hence confirming that changes in the b-value are associated with dynamic changes in the system rather than its initial conditions (i.e. initial stress distribution).

Fig. 3: Statistical analysis of the simulated seismicity for the step-increasing injection.
figure 3

a Temporal evolution of the b-value with a fixed time-window (0.2 days, moving average with a 0.1 days overlap; shaded area ± one standard deviation) for the homogeneous and fault cases; b Hourly a-value for the homogeneous and fault cases. For the homogeneous case: c Average spatial distribution of events over time; d Distribution of b-values during injection (light) and after shut-in (dark); the dashed lines do not include previously reactivated seeds; e Frequency-magnitude distribution of 200 events at 4 days (co-injection) and 8 days (after shut-in).

Full size image

For the homogeneous case, the overarching trend in the evolution of the b-value (and the a-value) shows a fast increase from the start of the injection followed by a plateau (if we disregard the pulses driven by repeating events). The shut-in or arrest of injection is followed by a decay of the b-value, which decreases to the natural state of the system. Figure 3d illustrates this natural state (dark) and the higher co-injection b-value distribution (light), driven up by repeating events. Using a T-test to evaluate the distributions of b-values co-injection and post-shut-in, we confirm the statistically significant difference of the b-value during and after the injection phase with a p-value of 1.95e-4. A case without allowing for previously stimulated regions to be reactivated, or with no stress changes, would result in a similar statistic as the natural post-shut-in (dashed lines in Fig. 3d). Figure 3e presents the frequency-magnitude distribution of two subsets of 200 events at 4 days (co-injection) and 8 days (post-shut-in), illustrating the influence of a changing b-value on the frequency-magnitude distribution.

Compared with the observed b-value in Basel51 seen in Fig. 1b, the homogeneous case in Fig. 3a does show a higher b-value during the injection with a decay following the shut-in, but in the observed data for Basel, the decrease in the b-value occurs two to three days before the shut-in happens. Such a decrease in the general b-valueb-value trend is observed in the fault case (Fig. 3a), suggesting that the observed behaviour in Basel is linked to the presence of the a-posteriori mapped fault53. This drop in b-value in Basel from day 3 has been correlated to a shift from a three-dimensional spread of the seismic cloud to a two-dimensional growth aligned with the main fault51.

This fault case displays a lower natural state of the system in the post-shut-in phase, due to the assumption that a fault zone has the potential for larger earthquakes (hence a lower b-value). The fault case shows the dip in the b-value once the fault starts being reactivated (at t = 1.3 days). This seems to tie in with the previous speculation made for Basel that the b-value trend changes once events align on the fault. It is worth mentioning that the synthetic fault simulated here does not get reactivated at a similar time to the one in Basel, as it is located closer to the injection point. The extremely similar behaviour of the a-value (Fig. 3b) between the homogeneous and fault cases strongly suggests that the b-value is the main driver of risk/hazard, since the number of occurring events does not diverge in time between the cases.

Influence of the site

The observed b-value varies widely between regions, thus has an impact on the risk (Fig. 1a). In Fig. 4, the same injection scenario is tested on different site conditions characterised by a local stress change from a more critically stressed site (corresponding to a lower intrinsic b-value here (lighter)) and a less stressed target site (with a higher intrinsic b-value (darker)). Figure 4a shows the evolution of the b-value for the base case and the two different state of stress cases, and Fig. 4b depicts the derived cumulative probability of a magnitude M ≥ 3.5 (as described in the Methods section). These cases show the importance of the initial state of stress of the target site in terms of risk. It is evident already that the regional b-value drives the hazard (and risk) and must be considered as the first-order element. However, the regional b-value is not easy to identify, especially in regions with low seismicity rates. Hence the importance of second-order variations, i.e. where the b-value changes with the injection, which could play an important role in identifying potential hazardous sites in the early stages of an injection.

Fig. 4: Influence of the regional state of stress on b-value and associated risk/hazard.
figure 4

a Evolution of the b-value over time with a ± 20% b-value variation around the base case (shaded area ± one standard deviation); b Corresponding cumulative probability of occurrence of an event of a magnitude ≥3.5.

Full size image

Injection strategies

While the risk is in general driven by local factors (i.e. low regional b-value and/or the presence of a fault), second-order effects on induced seismicity are clearly linked to the operational parameters (cumulative injected volume, injection rate, duration of injection)31,35,54,55. We devise three more injection strategies with the same total injected volume as the base case and test them using the same framework, both with and without a fault. Figure 5 presents the new injection strategies (linearly increasing, constant-rate and cyclical injections) alongside the base-case step-like injection. The onset of the fault reactivation (i.e. the average occurrence time of the first event occurring on the fault patch) is marked by a vertical dashed line, and the intrinsic b-value (i.e. the natural state of the system stabilised post-injection) as a horizontal dashed line for each injection strategy. We calibrate each model to reproduce a similar number of events to make the results comparable. Table 1 summarises the average number of events and the delta of the b-value between the co-injection and the natural state of the system for each injection strategy (thereby providing a quantitative overview of what Fig. 3d shows graphically for the homogeneous base case).

Fig. 5: Transient b-value for different injection strategies.
figure 5

For each panel, blue is the homogeneous case and green the fault case, the shaded areas show a ± one standard deviation interval, the horizontal dashed lines denote the intrinsic (i.e. at rest) b-value, and the vertical dashed lines show the earliest trigger on the fault. a Step increasing; b Linearly increasing; c Constant; d Cyclical.

Full size image

The constant injection scenario presents the lowest increase in the b-value during the injection, while strategies with a temporal increase in the injection rate (step- and linearly increasing scenarios) show much higher differences between the co-injection and post-shut-in b-values. Figure 5b, c show that the constant and linearly increasing strategies have the biggest differences between the homogeneous and fault cases.

Figure 5d shows that the cyclical injection makes it possible to get a preview of the natural state of the system post-shut-in early on in between the injection phases, similar to a test injection. This style of injection pattern would enable an early decision to be made on whether the risk associated with the project was worthwhile.

In Fig. 6, the cumulative probability of occurrence of a magnitude of at least 3.5 is presented for all injection strategies, for both without (solid lines) and with a fault (dashed lines). With the probability of occurrence Pocc we quantify the hazard posed by the strategies, and show that the fault cases are, as expected, riskier than the homogeneous cases due to their lower intrinsic b-value. We also see that the constant and cyclical injections are riskier than the increasing rates. The cyclical injection is distinguished from the others by clearly displaying three plateaus corresponding to the relaxation after the three injection phases. We did not consider the fatigue effect, which could substantially reduce the number of induced events (a-value) and hence result in a lower hazard.

Fig. 6: Cumulative probability of occurrence of an M3.5+ event for the different injection strategies.
figure 6

The solid lines correspond to the homogeneous cases, and the dashed lines to the cases with a fault.

Full size image

Operational injection scenarios’ influence on seismicity

Previously conducted sensitivity tests of different injection strategies and site-specific parameters on the seismicity and in particular on the simulated catalogue’s b-value concluded that the injection strategy does not have much of an impact on the overall b-value of the induced seismicity37. We find that the b-value averages over the whole simulation are indeed similar for all tested injection strategies, but that the b-value is dynamic and its evolution over time during the injection is largely influenced by the injection pattern (Fig. 5). Site-specific parameters (permeability, hydraulic properties of the host rock, density of fracture and stress conditions) have also been identified as having a wide-ranging effect on the simulated seismicity37. We show in the current study that the presence of a fault, which concentrates all of these site specificities, has a major impact on the behaviour of the seismicity and the b-value.

Cyclical stimulation treatments have been shown in hydro-fracturing cases to provoke mechanical and hydraulic fatigue of the rock mass by allowing for stress relaxation in between cycles56. So-called soft stimulation (repeated cyclical injections) tested in hydro-fracking settings yield better hydraulic performances despite their lower seismic efficiency by submitting the rock mass to fatigue due to repeated loading cycles, eventually resulting in fewer post-shut-in seismic events56,57,58. When tested in an EGS setting, such soft stimulations have had mixed results, with great successes in Helsinki59 but less desirable consequences when combined with other stimulation techniques in Pohang15,60.

Linking operational parameters like the injection rate and hazard/risk is a necessary step to go beyond comparative analysis of injection-induced seismicity. We propose using transient b-values as a diagnostic tool able to provide a robust risk assessment early on in prospective projects. In line with our conclusions, systematic injection tests (“Seismogenic Fault Injection Test”) have been proposed recently to assess the seismogenic index40, the b-value, and the decay rate of the post-shut-in seismicity34. Theses systematic injection tests are suggested to be conducted using a cyclical injection pattern to get repeated measurements of these target parameters. In the same context of linking the injection rate and hazard/risk, a recent study31 looked into the Basel case14 and tested alternative scenarios with a hybrid model to compare not only the direct output of the model but also the calculated risk posed by the alternative scenarios.

With our study, we show that, although secondary to site conditions, the chosen injection scenario does make a difference to the overall risk/hazard posed. However, such a risk/hazard must also be compared with the benefit of a stimulation. The estimation of the stimulated volume (i.e. the volume of the reservoir seeing a large permeability enhancement) is for now left to future endeavours, as our current model set-up only accounts for reversible pressure-dependent permeability changes. The TOUGH2-Seed framework does allow for the consideration of irreversible permeability changes (driven by slip38), and this aspect could be potentially explored in future work. Other important effects could be related to temperature as well as potential chemical effects.

Using the b-value for near-real-time hazard assessment and forecasting

Managing the risk posed by induced seismicity requires setting and enforcing regulatory measures to limit nuisance to the general public and potential damage. Traffic Light Systems have been widely adopted as risk mitigation method that is easy to implement and follow, with advantages and limitations discussed in recent years18. In general, three levels are set out in the protocol: green (where the operator is allowed to proceed unrestricted), yellow/orange/amber (where mitigation measures are required in response), and red (suspending all operations and potentially prompting bleed-offs and capping measures). The set limits for the thresholds have been based on recorded magnitudes, PGV or PGA (peak ground velocity or acceleration) depending on the project, and are usually calculated based on a preliminary risk assessment of the project. This simple regulatory control does not require there to be updates during the lifetime of the project/phase and remains limited in its efficiency. Monitoring the evolution of parameters important to the risk assessment (e.g. the a-value and the b-value) in near real time would make it possible to update the risk assessment and thresholds. Moreover, the use of numerical modelling in decision modules when determining thresholds and regulatory measures has the power to introduce game-changing technologies.

New approaches are being developed, where seismic data are fed into multiple models - from purely statistical to fully physical - which are then weighted and evaluated to provide robust ensemble forecasting of induced seismicity that can then be computed into hazard and operational control29,33,34,61,62. Such a chain from seismic data acquisition and geomechanical modelling to risk analysis and operational decision is integrated within an Adaptive Traffic Light System18,26.

The concept of Adaptive Traffic Light System has been tested both retrospectively on data from Basel61, and in real time during a well stimulation in Geldinganes (Iceland)32 and is currently being tested as part of the Bedretto lab63 and Forge Utah projects64. Multiple studies have assumed different scenarios of early injection arrest with statistical and simple hybrid models to produce pseudo-forecasts of the seismicity31,55,61,65. Such approaches did not look into changes in the b-value, despite recorded observations21,27. Although physical models can outperform statistical ones in probabilistic earthquake forecasting exercises66, the amount of computational time and power they require makes them unsuitable for near-real-time applications. Similarly, Epidemic Type Aftershock Sequences (ETAS) type models with dozens of parameters to invert at each time step are better suited to replays and pseudo-forecasting endeavours. For near-real-time applications, simplified models are preferred, and although they individually have their own shortcomings, ensemble modelling and weighting can provide a robust forecast32. Such models include variations of the Seismogenic Index type model40,41, modified approaches to account for spatial variability introduced by the geology67, or analytical solutions for simplified hybrid models31. Our proposal to monitor the b-value’s evolution over time could be used as a simple metric that provides fast-paced information with implications for risk.

Large numbers of events within highly complete catalogues extracted through template matching methods can increase the predictive power by tracking fine-scale changes in the spatial distribution and stress field66. In a similar way, we suggest that using the b-value evolution in near-real-time could allow us to detect deviations and changes in the stress conditions or even detect seismicity aligning on a fault before lineaments can be seen in the cloud of seismicity. This monitoring of the b-value, however, requires robustly generated magnitudes and generous catalogues recorded and generated in near-real-time to provide trustworthy signals from the underground68.

Maximum size of induced seismicity

In order to move towards safer industrial underground operation, one needs to understand what governs the size of induced earthquakes. For a given regional tectonic context, the injected volume seems a logical first-order control on induced seismicity, and observations tend to agree in wastewater disposal5 and hydro-fracturing67 cases. Some models based on a direct control of the volume on the number of events have been proposed41 and used in near-real-time32 or to introduce risk informed thresholds29. The injected volume can play a tremendous role on the size of the seismicity (Supplementary Fig. S7). In this study, we chose to look beyond this first-order control with constant injection rates for all the cases presented in this study. Indeed, the example of Pohang (2017, South Korea) demonstrates that there is more than just the volume at play, with only a fraction of the expected needed volume injected for an earthquake of magnitude Mw 5.515,69. So far, attempts at binding the maximum magnitude of operations have gone in two directions: on the one hand focusing only on the volumes or rates of fluid70,71, and on the other hand looking into the seismicity catalogue statistics72. Both approaches give broad estimates for the range of the maximum magnitude and could provide valuable near-real-time insights by taking into account temporal changes of the b-value and injection volumes during operations. Other models are being developed to take into account the physics of an arresting rupture73, frictional behaviours, and the evolution of the injection rate74. In terms of quantifying the size of an induced seismicity sequence, statistical laws give us some clues as to how and when a sequence stops. Both Omori and Båth’s laws for aftershock decays have been adapted to induced seismicity, as well as other statistical decay rates34,75. A recent study found that a modified Båth’s law accounting for injected volume can estimate the statistical decay of induced sequences34. The shut-in phase allowing for estimation of decay times and investigating diverse shut-in patterns would constitute an important next step. The influence of the stress conditions (and thus the b-value) on the nucleation and decay times has also been highlighted76, suggesting once again the paramount importance of monitoring the b-value evolution for a careful risk assessment.

Relationship between b-value and stress

As shown in laboratory experiments42,43,44, the Gutenberg-Richter b-value seems to be correlated with the stress conditions in the medium. Observations of a relationship between b-value and pore pressure in induced seismicity sequences show a direct proportionality in the Denver earthquake sequence77, and this is supported by a b-value for high-pressure injections in geothermal reservoirs that is higher than the regional average28. The correlation between the stress regime and the b-value suggests a control of the faulting regime by the differential stress (and thus by the confining pressure)45,46. The b-value has been interpreted as a stress-metre in the Earth’s crust, and heterogeneities in the mapping and evolution of the b-value must be taken into account when performing hazard or aftershock analysis50. The inverse relationship between the b-value and differential stress has been observed at all scales, from laboratory samples to mega-ruptures, and holds true across stress regimes, regions and depth ranges46. This scale invariance and permanency in the relationship between the b-value and stress suggests that transient b-values can only be explained by long-term changes in the stress state of the medium, as observed after a major earthquake49. Transient changes in the stress field constitute a second-order effect, and, while important to consider, they do not fully control the hazard. As we have shown in Fig. 4, the site-specific stress conditions play a major role in the intrinsic risk.

Recently, a unified solution has been proposed to explain both the depth and stress regime dependencies, going as far as to propose a global analytical relationship between differential stress and the b-value46,52. We approach the modelling of the b-value from this global context, adapting the analytical solution to fit observations made in Basel51. The ability of our model with fault to reproduce both the initial increase and then the decrease in the b-valueb-value days before shut-in suggest that the observations made in Basel can be explained by the injection strategy and the presence of a fault. Furthermore, the agreement between the observations from Basel and our model reinforces the importance of accounting for a relationship between the b-value and stress in general for a better understanding of the physics behind induced seismicity sequences. This direct relationship between stress and the b-value highlights the importance of monitoring the b-value before, during, and after stimulation operations to use it as a risk-metre. However, no direct observation has yet provided conclusive evidence, as stress measurements in real-time remain out of reach.

Conclusion

In this study we use a hybrid modelling approach - TOUGH2-Seed - to simulate stimulation injections into tight rock in an effort to approximate an EGS, and provide a hypothesis explaining the observed transient b-value in Basel51. We use the Gutenberg-Richter power-law parameters as a transient proxy for risk. In this we go beyond traditionally used statistical models in providing simulation results with both the number of events but also their power-law distribution. We show that the choice of injection strategies impacts the transient b-value and thus risk during the injection phase. Furthermore, we show in a quantitative and objective way that the presence of a hydraulic and seismogenic fault plays a major role in the hazard/risk associated with injection operations. In parallel to event localisation, monitoring the evolution of the b-value over time may enable detection of such a fault prior to major events and could lead to operational techniques noticing a deviation from an expected b-value pattern. Finally, the Gutenberg-Richter b-value could provide an image of permeability changes in the subsurface, given that more permeability changes accommodate and allow more pressure build-up, as seen when comparing our homogeneous and fault cases. The presence of a more permeable feature such as a fault can indeed be detected early on with a decrease in the general b-value trend or a deviation from the expected b-value trend. Real-time monitoring of the b-value, coupled with rapid seismic relocation techniques, could help to detect changes in the hydraulic response of the system (i.e. permeability changes leading to pathway creation mapped through seismicity lineaments and cloud geometry changes detectable in decreasing b-values).

Methods

TOUGH2-Seed

TOUGH2 is a commercial software providing full 3D, multi-phase and multi-component flow simulation78. We couple TOUGH2 to a 3D geomechanical-stochastic seed model28,55. The approach accounts for 3D stress fields and allows us to distribute potential hypocentres (with strike, dip, and rake) in the medium. Once coupled, the hybrid TOUGH2-Seed model38 allows for reversible permeability changes dependent on pore pressure variations. TOUGH2-Seed relies on a Mohr-Coulomb failure criterion evaluating seed reactivation at each time step: The effective normal stress (\({\sigma }_{n}^{eff}\)) on the seeds is changed by the pore pressure, if this this change of \({\sigma }_{n}^{eff}\) is enough to exceed the critical shear stress (referred to as shear strength), the seed is reactivated. Additional physical phenomena (e.g. irreversible shear-based permeability enhancement and earthquake-earthquake interaction through Coulomb static stress transfer) are available in TOUGH2-Seed but not used in this study. More details about the various physical mechanisms available can be found in Rinaldi & Nespoli, 201738. TOUGH2-Seed38,79 has shown promising capabilities for different geothermal settings from hydrothermal field scale (Hengill, Iceland79) to complex hydrothermal systems involving a gas phase (Sankt Gallen, Switzerland30), and EGS (Basel, Switzerland38).

Our hybrid model is made up of a deterministic TOUGH2 part which is run beforehand on a single core machine to get a pressure and temperature solution; and of a stochastic Seed part coded in MATLAB. We run the model uncoupled from the pressure solution as a static input to the Seed model. This code is run on an LFS (Load Sharing Facility) cluster for 48 realisations on 48 cores and takes roughly 30 min of parallel computation time for a 10-days simulation with ~500 time steps and 210.000 seeds. We used 48 realisations given that a larger number of simulations will result in a similar statistical variation in the b-value (see Supplementary Fig. S9).

Assignment of the b-value to seeds

The geomechanical Seed model contains a full 3D formulation of the stress tensor, and each seed is thus assigned an initial stress condition. Both lab and field evidence shows a linear relationship between differential stress and the b-value. We use a linear relationship similar to the one proposed by Scholz52, adapted to fit the amplitude of the b-value observed in Basel (Fig. 1b51):

$$b=2.8-0.02{\sigma }_{diff}$$
(1)

Equation (1) was calibrated using the maximum amplitude of the b-value observed in Basel and fitted to the range of differential stress in the modelled reservoir. Supplementary Figs. S4 and S5 show the spatial distribution of the change in b-value for the base and fault cases respectively.

Stress drop and differential stress modelling

The stress conditions on any given seed can be modified by pore pressure and reactivation. Upon reactivation of a seed, a stress drop is applied and the stress tensor on the seed recalculated, allowing the b-value of this seed to change. The applied stress drop is calculated to be a fraction of the normal stress on the seed (see the stress-drop factor in Table 2). The stress drop is also distributed in the calculated rupture area of the initially reactivated seed, allowing neighbouring seeds to also see their stress conditions, and thus their b-value change. These changes in stress drive the changes in the b-value, in particular when a seed gets reactivated and subjected to a stress drop more than once, leading to its differential stress decreasing and its intrinsic b-value increasing. The assumption of a stress drop distributed over the rupture area or punctual only on the seed does not affect the overall earthquake statistic (see Supplementary Fig. S10).

Table 2 Hydraulic and physical parameters used in the TOUGH2-Seed simulations.
Full size table

In Supplementary Fig. S10, we test two approaches to model the stress drop: one local applied only to the reactivated seed, and a distributed stress drop that affects neighbouring seeds. This second approach aims at simulating a rupture area for each event. The extent of the rupture area is based on the average expected magnitude from the power law distribution with the b-value of the given seed. Both approaches give very similar results for comparable cases (with/without a hydraulic fault). The distributed approach is more realistic and allows us to release the stress in a region around the given event. In the distributed case, the stress drop occurs within a radius defined by the magnitude of the reactivated seed. Equation (2) details the calculation of the rupture area according to Brune’s model, with M0 the seismic moment calculated from the picked magnitude (Eq. (3)), and Δτ is the stress drop applied to the reactivated seed.

$${R}_{eq}=\sqrt{\frac{7}{16}\frac{{M}_{0}}{\Delta \tau }}$$
(2)
$${M}_{0}=1{0}^{\frac{3}{2}M+9.1}$$
(3)

Setup and parameters of the model - base case

The model comprises a 4-km-deep injection placed in a 1.5 × 1.5 km reservoir with 2 km of thickness centred around the injection point. The injector is placed in the centre of the volume. The mesh consists of Cartesian blocks of 100 m by 100 m by 100 m and is refined in the centre to 25 m × 25 m horizontally and 50 m in depth. The stress and fluid conditions are selected to be as generic as possible, making use of the average crust model for stress conditions27; a summary of the principal simulation parameters can be found in Table 2. Cold water is injected for 5 days, with varying rates for the different injection strategies (inlets in Fig. 5).

Setup and parameters of the model - fault case

To the base model a vertical-dipping hydraulic fault striking 155∘N is added (to match the observed fault reactivated in Basel). The total number of seeds in the domain is kept identical (at 210,000 seeds) with a subset of 20,000 seeds now distributed on the permeable fault. The fault case comprises multiple layers of complexity, with a hydraulic expression in the TOUGH2 model, a specific population of seeds having a characteristically lower b-value.

The hydraulically conductive fault zone is added to the homogeneous case by setting a higher permeability (1.0 ⋅ 10−15m2) to specified cells in the TOUGH2 mesh. These cells fall within a 1 km by 1 km vertical plane, with 25 m of thickness located at a distance (100 m) from the injection. The pressure distribution in the system is changed by this high permeability feature (see Fig. 2c, e for comparison). All the steps in the fault build-up are documented in Supplementary Fig. S11.1, with the number of events are b-value drop reported in Supplementary Table S2. The influence of the fault permeability on the transient b-value is shown in Supplementary Fig. S11.2.

The fault zone is assigned a population of seeds with a higher density than the rest of the domain (20,000 seeds on the fault, 190,000 in the homogeneous domain). Seed strike orientation and fault orientation/stress rotation so that seeds on the fault are optimally oriented for reactivation in a Mohr-Coulomb style failure domain. Faults tend to have different rupture behaviours from the host rock. This is mostly seen as faults having a different stress state from the surrounding intact rock. The stress tensor on the fault cannot be changed directly as this would introduce a discontinuity in the stress field into our model. To simulate this fault-specific state of stress, we use the b-value (direct proxy for differential stress) and coefficient of friction of the seeds contained in the fault zone. Thus, the fault zone is made more seismogenic than the rest of the domain in two steps. First, the initial b-value of the seeds on the fault is lowered by 20% compared with the host-rock seeds (increasing the likelihood of larger events). This lowering of the b-value for highly clustered seeds naturally decreases the overall b-value. Second, the coefficient of friction of the fault’s seeds is lowered by 20% to weaken the fault (making the seeds on the fault easier to reactivate and allowing seeds further from failure to get reactivated). This last step has the twin effect of not only reactivating newer or more intact seeds but also resulting in more repeater seeds, hence the relative increase in the b-value compared with the previous step. Supplementary Fig. S11.3 also presents the case of a fault located further from injection (at 200 m).

For the fault case, we chose the hydraulic fault with lowered b-value located 100 m from the injection.

Calculation of a- and b-value evolution in time

For each realisation, the b-value is averaged in bins of 0.2 days (moving window with a 0.1-day increment). Each realisation’s time series can be seen with each thin line in Supplementary Fig. S6-a. The time series of b-values are then averaged over the 48 realisations to compute the preferred b-value (see the thick line in Supplementary Fig. S6a, and analysis of the computation of the uncertainty margin in Supplementary Fig. S12). The a-value is calculated in the same time intervals and normalised to obtain the hourly a-value (see Fig. S6b).

Calculation of probability of occurrence

For each realisation, the b-value is averaged in bins of 0.2 days, without any overlap, and the a-value corresponding to that time window is normalised to an hourly a-value. The time of recurrence (Tr) and the probability of occurrence (Pocc) for an event of magnitude M or above, is then calculated using the normalised a-value and the time series of the b-value with Equation (4) and Equation (5)48. The final probability of occurrence is then averaged over the 48 realisations (see the thick line in Supplementary Fig. S6c) and summed up to get the cumulative probability of occurrence (Supplementary Fig. S6d).

$${T}_{r}(M)=\frac{1}{1{0}^{a-bM}}$$
(4)
$${P}_{occ}(M)=1-{e}^{\frac{-1}{{T}_{r}(M)}}$$
(5)

Supplementary Fig. S8 presents the extension of Fig. 6 to higher magnitudes.