Induced seismicity closed-form traffic light system for actuarial decision-making during deep fluid injections

Abstract

The rise in the frequency of anthropogenic earthquakes due to deep fluid injections is posing serious economic, societal, and legal challenges to many geo-energy and waste-disposal projects. Existing tools to assess such problems are still inherently heuristic and mostly based on expert elicitation (so-called clinical judgment). We propose, as a complementary approach, an adaptive traffic light system (ATLS) that is function of a statistical model of induced seismicity. It offers an actuarial judgement of the risk, which is based on a mapping between earthquake magnitude and risk. Using data from six underground reservoir stimulation experiments, mostly from Enhanced Geothermal Systems, we illustrate how such a data-driven adaptive forecasting system could guarantee a risk-based safety target. The proposed model, which includes a linear relationship between seismicity rate and flow rate, as well as a normal diffusion process for post-injection, is first confirmed to be representative of the data. Being integrable, the model yields a closed-form ATLS solution that is both transparent and robust. Although simulations verify that the safety target is consistently ensured when the ATLS is applied, the model from which simulations are generated is validated on a limited dataset, hence still requiring further tests in additional fluid injection environments.

Full text

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SCIEnTIFIC RePoRTs | 7: 13607 | DOI:10.1038/s41598-017-13585-9
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Induced seismicity closed-form
trac light system for actuarial
decision-making during deep uid
injections
A. Mignan
1,2,3, M. Broccardo
2, S. Wiemer3 & D. Giardini1
The rise in the frequency of anthropogenic earthquakes due to deep uid injections is posing serious
economic, societal, and legal challenges to many geo-energy and waste-disposal projects. Existing
tools to assess such problems are still inherently heuristic and mostly based on expert elicitation (so-
called clinical judgment). We propose, as a complementary approach, an adaptive trac light system
(ATLS) that is function of a statistical model of induced seismicity. It oers an actuarial judgement of
the risk, which is based on a mapping between earthquake magnitude and risk. Using data from six
underground reservoir stimulation experiments, mostly from Enhanced Geothermal Systems, we
illustrate how such a data-driven adaptive forecasting system could guarantee a risk-based safety
target. The proposed model, which includes a linear relationship between seismicity rate and ow
rate, as well as a normal diusion process for post-injection, is rst conrmed to be representative of
the data. Being integrable, the model yields a closed-form ATLS solution that is both transparent and
robust. Although simulations verify that the safety target is consistently ensured when the ATLS is
applied, the model from which simulations are generated is validated on a limited dataset, hence still
requiring further tests in additional uid injection environments.
A signicant proportion of the world’s global energy production relies on subsurface resources, such as oil, gas
and coal production, as well as geothermal energy. In addition, the deep underground is increasingly used for
waste storage; typical examples are wastewater from fracking operations and CO2 sequestration. However, these
technologies are not “risk-free,” as shown by the increased frequency of induced seismicity cases around the
globe. Recent examples include induced seismicity related to fracking and wastewater disposal1–3, gas extraction4,
gas storage5, CO2 sequestration6, and renewable geo-energy7–9. Although some jurisdictions have enforced the use
of maximum magnitude thresholds to limit the induced seismicity risk10, most of these rules remain heuristic. In
this article, we argue that quantitative risk assessment and mitigation strategies rather than trial-and-error meth-
ods, should be essential tools to guarantee safety for society. e approach we advocate, which should be seen as a
proof-of-concept, allows for an informed risk-cost-benet analysis involving all stakeholders11.
Trac light systems (TLS) are commonly used to mitigate induced seismicity risk by modifying the uid
injection prole1,10,12,13. A TLS is based on a decision variable (earthquake magnitude, peak ground velocity,
etc.) and a threshold value above which actions (e.g. stopping the injection or reducing production rates) must
be taken. Currently, the denition of this threshold is based on expert judgment and regulations10,12,13. Here, we
propose a data-driven adaptive TLS, termed ATLS, which aims to overcome the limitations of the traditional
heuristic methods. Here, the assignment of a magnitude threshold is based on a quantitative risk assessment,
subject to a safety criterion imposed by the authorities (e.g., xed probabilities of unaccepted nuisance, damage or
fatalities). As a consequence, the ATLS is an objective and statistically robust mitigation strategy, which facilitates
a fair and transparent regulatory process. is approach is in line with the procedures common for most other
technological risks, such as in the hydropower, nuclear or chemical industries14. Model-based forecasting and
alerting are already advocated elsewhere, such as in hurricane data assimilation and forecasting15.
1Swiss Federal Institute of Technology Zurich, Institute of Geophysics, Zurich, Switzerland. 2Swiss Competence
Center for Energy Research – Supply of Electricity, Zurich, Switzerland. 3Swiss Seismological Service, Zurich,
Switzerland. Correspondence and requests for materials should be addressed to A.M. (email: arnaud.mignan@sed.
ethz.ch)
Received: 4 July 2017
Accepted: 25 September 2017
Published: xx xx xxxx
OPEN
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Results
Predictive hazard model. A predictive model lies at the heart of any risk assessment. In the case of induced
seismicity, a wide range of statistical and physics-based models exists16,17. Undoubtedly, more work is needed to
develop, calibrate and validate new models; however, we believe that the missing link is the use of such models for
deriving and monitoring quantitative risk thresholds. Here, we use as example a simple and yet robust model that
forecasts the piecewise induced seismicity temporal, here daily, rate λ(t, m ≥ m0; θ) as:


λ
τ
≥θ=











≤



−−



>
−−
−−−−
tm mVt tt
Vt tt tt
(, ;) 10 () ;
10 ()exp;
(1)
abmshut in
abmshut in shut in shut in
0
fb
fb
0
0
where
V

(t) is the injection ow rate as a function of time t in m3/day, θ = [b, a, τ] a set of model parameters
describing the underground characteristics (earthquake size ratio, activation feedback in m−3 and mean relaxa-
tion time in days, respectively), m0 the minimum magnitude cuto, and tshut-in the shut-in time, also in days. Both
a and b can also be functions of time t (see Decision variable section). In this model, the injection or operation
phase is described by a linear relationship between λ(t, m ≥ m0) and

V
(t) in line with previous studies17–19. It
derives directly from the linear relationship between

V
and overpressure17, hence assuming no change of injectiv-
ity during any given stimulation. e post-injection phase is described by a pure exponential decay representative
of a normal diusion process17. Although the Modied Omori Law is sometimes used to describe post-injection
seismicity20, reasons remain mostly historical21,22. e proposed alternative is veried to be consistent with the
tested data (see Methods section) and preferred on analytical grounds, being directly integrable in contrast with
the Modied Omori Law, which is conditional on parameter values and may require the denition of an ad hoc
upper bound22. Finally, no maximum magnitude Mmax is imposed. It follows from Eq. (1) that induced seismicity
is characterized by both the injection prole

Vt()
, and the underground feedback described by the three-parameter
set θ. e main limitations of the proposed model are discussed in detail later on.
In contrast with complex uid modelling16, the closed-form Eq. (1) can be computed on-the-y; moreover,
it includes the mean relaxation time, τ, hence taking into account the long-term underground feedback aer
shut-in. Finally, being integrable, it leads in turn to a closed-form ATLS, as demonstrated in the next section.
Although the physical process governing the rate of induced seismicity is more complex than what is repre-
sented by Eq. (1), this rate model is proven to be valid in a Poissonian probabilistic setting (see Methods section).
Moreover, the physical processes are either not still clear (in fact, there is not an unanimous consensus among
scientists), or computationally expensive. erefore, pragmatism imposes the use of statistical models until both
an agreement is found on the physics of induced seismicity and computational time of complex physical model-
ling is reduced.
e model (Eq. 1) was tted to six induced seismicity sequences observed in uid injection experiments from
enhanced geothermal systems (EGS)13,23–26, the initial stage of a long-term brine sequestration27, and one fracking
at an oil eld28 (Table1). e model succeeds to describe most of the data as shown in Figure1 (see results of sta-
tistical tests in the Methods section – Note that the rare outliers above 3σ may be due to missing on-site data that
may aect seismicity, such as unknown technical operations on wells, or to second-order physical processes miss-
ing in Eq.(1), as discussed below). e parameters are found to range over 0.8 ≤ b ≤ 1.6, −2.8 ≤ a ≤ 0.1 m−3 and
0.2 ≤ τ ≤ 20 days and show a relatively large scattering between sites and between dierent stimulations at a same
site (Table2). It is important to note that we specically chose those six datasets, as they had been made publicly
available. ose cases are characterized by high pressures and ow rates, and are rich in induced earthquakes.
is represents a selection bias and the a parameter could decrease to much lower values elsewhere (Fig.2). For
instance, most injection wells in the U.S. do not cause felt earthquakes29; large a variations between regions and
sites might be explained by dierent regional crustal stresses30. e present sites provide however natural labo-
ratories to test our model and the associated ATLS, without generalizing or inferring any high level of risk for all
existing deep uid injections. Figure2 illustrates the parameters’ scattering, including results from past studies18
for EGS and other hard rock settings enlarging the range of values to 0.7 ≤ b ≤ 2.2 and −4.2 ≤ a ≤ 0.4 m−3. It has
been shown that in fracking environments, the activation feedback can be as low as a = −9.25 m−3 18.
Site (country ISO
code), year Injection prol e Earthquake catalog
Basel (CH), 2006 Digitized from (13) (23)
Garvin (US), 2011 Digitized from (28) (28)
KTB (DE), 1994 Digitized from (24) (24)
Paradox Valley
(US), 1994 http://www.usbr.gov/uc/wcao/progact/paradox/RI.html http://www.usbr.gov/uc/wcao/progact/paradox/RI.html
Newberry (US),
2012 Digitized from (25) http://fracture.lbl.gov/cgi-bin/Web_CatalogSearch.py
Newberry (US),
2014 Digitized from (26) http://fracture.lbl.gov/cgi-bin/Web_CatalogSearch.py
Table 1. Source* of stimulation experiment datasets. *Online material last assessed in June 2017.
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Safety criterion. A safety criterion is a probability of exceedance that can be xed with respect to dierent
safety metrics, such as fatalities, economic loss, building damage or level of nuisance14. Given the selected metric,
the corresponding safety criterion can be converted in the magnitude space into the probability of exceedance
Pr(m ≥ msaf) = Y, which will ensure that the acceptable level of risk is preserved at all time:
Figure 1. Induced seismicity model tting of six uid injection experiments: 1994 German Continental Deep
Drilling Program (KTB); 1994 Paradox Valley, United States; 2006 Basel, Switzerland; 2011 Garvin, United
States; and 2012–2014 Newberry, United States. For both KTB and 2014 Newberry, experiments are broken
down into two separate stimulations, each with its own post-injection tail. e model (Eq.1) is represented
by the red curves on the induced seismicity time series with the ±3σ uncertainty envelope shown in light red.
Dashed and dotted vertical lines indicate the shut-in times and sub-stimulation periods, respectively.
Experiment m0b a (m−3)Σ*τ (days)
B06 0.8 1.58 0.10 0.4 1.12
G11 1.0 0.77 −1.35 N/A 0.28
KTB94a −1.5 0.98 −1.35 −1.65 0.03†
KTB94b −1.4 0.87 −1.65 −1.65 0.22
PV94 0.6 1.08 −2.40 −2.6 14.13
NB12 0.2 0.80 −2.80 N/A 12.59
NB14a 0.0 0.98 −1.60 N/A 3.55
NB14b 0.2 1.05 −1.60 N/A 3.16
Table 2. Maximum likelihood estimates per stimulation experiment. *Seismogenic index obtained by ref.18;
†unreliable, part of the tail being likely hidden by the KTB94b sequence.
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τ≥=−− +=
−
−−

()
mm Vt Vt YPr 1exp{10[()()]}
(2)
saf
abm
shut in shut in
fb saf
with msaf the magnitudeat which the given safety limit (e.g., damage, fatality) is reached. Note that Eq. (2) is a
closed-form expression where V(t) is the cumulative injected uid volume, and msaf and Y are derived from the
safety criterion (see Methods section). Moreover, the set of parameters θ is updatable at any given time. e map-
ping from risk to earthquake magnitude is required to control injection operations based on short-term observa-
tions (see Decision Variable section). While peak ground velocity (PGV) is a more direct measure12, conversion to
magnitude is in any case inevitable to estimate the risk potential of larger earthquakes from the b-value. It is also a
unique measure, while PGV requires a location that is not trivial to assign. It should be added that no maximum
magnitude Mmax is imposed in Eqs (1–2). is implicitly assumes that both small to medium-size induced events
and large triggered earthquakes on existing faults are treated the same way. is remains debated31 although a
recent study19 demonstrated that the observed Mmax in uid injections is compatible with the null-hypothesis
of the Gutenberg-Richter law with no upper limit. e role of Mmax (and therefore of triggered earthquakes) is
however only critical when the risk of fatalities (e.g., individual risk IR) is evaluated. For nuisance or minor dam-
age thresholds, risk is more likely dominated by medium-size induced events. is important discussion has no
signicant impact on the method proposed, as proved in the Methods section.
Before an ATLS is set, the likelihood of failure of the planned project with respect to a specied limit state
function dened by the safety criterion in magnitude space (Eq.2) can approximately be determined. In this
study, we select as main metric the annual individual risk (IR) over the entire project period, and as safety crite-
rion IR ≤ 10−6 (i.e. the probability that a statistically representative individual dies for the introduced hazard),
Figure 2. ree-parameter set θ = [b, a, τ] scattering in uid injection experiments & impact on project
validation for xed safety threshold (IR ≤ 10−6) and dierent uid injection scenarios. Dots represent θ
estimates obtained in the present study (Fig.1), squares the ones obtained by ref.18 (with no information on τ)
and curves (made of successive dots) time-dependent estimates obtained where the earthquake catalogue is
large enough (see Methods section). e hypothetical project injects a total volume V = 10,000 m3 of uids with
constant ow rate

V
at a distance d from the nearest building: a.
V

= 1 m3/min, d = 0 km; b.

V
= 1 m3/min,
d = 50 km; c.
V

= 10 m3/min, d = 0 km and d.

V
= 10 m3/min, d = 50 km. Lines represent the safety threshold for
dierent values of τ (in days) (Eq.2). Colour green or red indicates if the safety target is respected or not,
respectively (darker colours are used in the case of time-dependent estimates). Letters represent sites: Basel (B),
Cooper Basin (CB), Garvin (G), KTB, Newberry (NB), Ogachi (O), Paradox Valley (PV) and Soultz (S),
followed by the experiment year’s last two digits.
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which is a threshold commonly enforced for hazardous installations14. Figure2 shows the acceptable domain for
a xed limit state function, tested for dierent injection scenarios in which a hypothetical project plan is to inject
a total volume V = 10,000 m3 of uids at a depth of 4 km32. e injection prole is assumed to be at with a con-
stant ow rate
V

= 1 or 10 m3/min, having an impact on injection duration and tail behaviour (Eq.2; Fig.2). e
project is considered to be located at a distance d = 0 km or 50 km from the nearest building. For a given site, with
no knowledge of the underground feedback to uid injection, project operators and regulators in an EGS setting
(where high seismicity rates are common during stimulation) could use the known θ scattering for an a priori
parameterization. is preliminary assessment shows the likelihood of the project to pass or fail the safety thresh-
old. As shown in Figure2, the results can be ambiguous, due to the large uncertainties associated with subsurface
characteristics. Nevertheless, it provides a preliminary assessment of the risk reecting the limited knowledge of
the induced seismicity process. Future estimations of θ at additional injection sites will likely rene the results and
improve the decision process. In addition, rules of decision-making under uncertainty can account for that ambi-
guity9,33. Decisions become more obvious in cases in which the diagram would be entirely green (clear go) or red
(clear no-go). Underground stimulation activities in areas with low exposure (e.g. remote EGS plant locations
with large distance d from the nearest habitations) would evidently have a lower induced-seismic risk and, thus,
shrink the red area. e termination of the 2006 Basel EGS project was due to the high induced-seismic risk
emerged from the high exposure of the urban built environment7,9. Note that Eq. (2) can be used to predetermine
a distance d for which the induced-seismic risk would become acceptable—conditional to a given injection prole
V(t) and parameter set θ (since msaf is a function of d; see Methods section).
Decision variable. e ATLS decision variable must be selected and updated as new data allows estimating
θ more accurately, or if the planned injection scheme is changed. Here, a threshold earthquake magnitude mth is
used as decision variable. In particular, mth is dened as the magnitude value for which mitigating actions must
be taken, here corresponding to stopping injection, i.e.

τ=− +
−−
m
b
YVtm
1
log[ 10 ()]
(3)
th abmshut in saf
10 fb saf
(see Methods section). If mth is updated “on-the-y”, the project is guaranteed to meet the dened safety criterion.
To avoid reaching mth before the planned stop of the uid injection one may be inclined to reduce the ow rate
V

, but here it would only delay the time at which the injection must stop, as the risk is mostly controlled by total
volume injected V (Eq.2; the secondary role of

V
is highlighted in Figure2 by the change of width of the yellow
band, for dierent

V
and τ). It is common practice to reduce
V

however13, but results of such action remain
unclear34 since verication of a safety threshold requests a large number of experiments (see simulations below);
indeed: (i) It is plausible that such action has no overall eect, the risk remaining the same in average over a xed
V; (ii) If such action has an eect, it would indicate that the model of Eq. (1) does not properly describe the role
of dierent injection strategies. Despite the proposed model being classied as a pure statistical method, it is
based on physical considerations. Its rst term, for example, builds on the linear relationship between volume
change and pressure17. Other relationships can be envisioned such as a bilinear relationship indicative of a change
of injectivity26, which may explain some second-order relationship observed between ow rate and Mmax35. e
linear relationship could also be shied in time by including a minimum pressure threshold17 below which no
induced seismicity is triggered. Adding such processes would likely allow for smarter mitigation strategies in
which the shape of the injection prole would play a role. Since any model change would require the inclusion of
additional parameters (which have yet to be constrained), and since Eq. (1) is veried to be consistent with most
of the data tested, we consider Eq. (1) to be a reasonable rst-order model for the proposed ATLS.
To validate the ATLS in a realistic time-dependent setting, we simulate the Basel induced seismicity sequence
using Eq. (1) with

V
Basel(t) (Figs1; 3a) and θBasel(t) = [b(t), a(t), τ = 1.12 day] (Figs2; 3b; Table2). We use the
safety threshold IR ≤ 10−6 with the nearest building above the borehole (d = 0 km), y ielding msaf = 5.8 (see
Methods section). e value mth as a function of time is shown in Figure3c with two examples of induced seis-
micity time series. e rst one, in grey, is a reproduction of the 2006 Basel experiment; the second one, coloured,
is the case where the ATLS is used. Figure3d nally shows that the safety target, while not respected for synthetic
versions of the Basel experiment, is reached once the newly proposed ATLS (Eq.3) is considered. Due to the
stochastic nature of the earthquake process, the operational safety target is only reached on average over multiple
sequences. It is plausible that an improved model with a non-linear relationship between ow rate and overpres-
sure would open the possibility for mitigation strategies based on dierent injection proles, hence potentially
avoiding prematurely stopping the stimulation and the project.
Discussion
e main purpose of this study was to present a statistical-based rst order ATLS, which veries that a quan-
titative safety criterion is ensured. An important benet of the outlined ATLS approach to both operators and
regulators is its transparency and execution speed (being a suite of simple closed-form expressions). By princi-
ple, its adoption would make any project in compliance with the safety threshold whatever the response of the
underground. is does not ensure that a project is nancially successful, but it gives the operator the maximum
allowable chance to reach success, based on a quantitative risk-based method.
It is important to note that existing trac light systems based on heuristics can already provide reasonable
results (for instance, the magnitude threshold mth = 2.9 xed during the Basel experiment13 is not far o the
mth(t) computed with the new ATLS; Fig.3c). However, magnitude thresholds enforced by dierent jurisdictions
vary signicantly (with 0.5 ≤ mth ≤ 410) with no clear link with the standard risk-based safety criteria used in
other hazardous industries14. Although the application of the ATLS in the decision process might appear at rst
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SCIEnTIFIC RePoRTs | 7: 13607 | DOI:10.1038/s41598-017-13585-9
complex, there is substantial evidence that the algorithmic (or actuarial) approach is superior to the so-called
clinical approach of informal judgement36,37. is must apply too to induced seismicity prognostics, experts nec-
essarily basing their judgements, consciously or intuitively, on past observations shown here to be reasonably well
described by Eq. (1) (Fig.1 and Methods section).
Finally, we suggest the following future directions: (1) improve the model (Eq. 1) by relating directly over-
pressure instead of ow rate to induced seismicity. Due to potential changes in injectivity, gas kicks and other
processes, overpressure is likely to provide a better proxy than injected uid volume; (2) test the statistical model
in other uid injection environments; (3) improve the updating of the parameter space by using a hierarchical
Bayesian framework where the uncertainties of the model parameters are taken into account; and (4) modify
the mapping from risk to magnitude space for site-specic conditions which likely vary between uid injection
locations. e present study demonstrated the power of the actuarial approach and should be considered as a
proof-of-concept for future physics-based induced seismicity models, more sophisticated engineer-based risk
assessments, and improved mitigation strategies.
Methods
Time series analysis. e induced seismicity temporal rate model λ(t) of Eq. (1) is tted to the induced seis-
micity data by using the maximum likelihood estimation (MLE) method38. e probability pi that the observed
number ni of induced earthquakes results from a Poisson process39 with rate λi is
p
n
exp( )
!(4)
ii
n
i
i
i
λλ
=
−
which yields the log-likelihood function
Figure 3. ATLS validation on synthetics of the 2006 Basel uid injection experiment. (a) Observed time series
and ow rate

Vt()
during the 2006 Basel experiment13,23; (b) Maximum likelihood estimates (MLE) of ground
parameters a and b over time determined from the Basel time series using a moving window of 100 events (see
Methods section). Generic parameter values are here updated aer the rst 100 events are observed. Once
injection stops at tshut-in, b is assumed to remain constant since no control on induced seismicity is possible aer
that time; (c) Simulated version of the 2006 Basel time series (in grey) and shorten (coloured) when injection is
stopped by having an event with magnitude m > mth (in green); (d) ATLS validation showing that the frequency
of events with m ≥ mth tends to Y over many simulations (uctuations around Y represent the inherent
uncertainty of the earthquake Poisson process, which decrease with increasing sample size).
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LL Xpnn(, )ln( )[ln() ln(!)]
(5)
i
imax
i
i
imax
ii ii
1
∑∑
θλλ== −−
=
where X = {n1, …, ni, …, nimax} is the observation set and θ = [a, b, τ] is the parameter set of Eq. (1). e maxi-
mum likelihood estimate of θ is nally θMLE = arg maxθ LL(θ, X). m0 is xed to Mc, the completeness magnitude
dened as the magnitude bin with the highest number of events40. b is estimated independently of λ, also based
on the MLE method41. e model is tted to 8 datasets (from 6 stimulations in various injection settings; Table1);
the resulting maximum likelihood estimates are listed in Table2.
Sensitivity analysis. Temporal changes in a and b are evaluated for induced seismicity sequences that are
large enough (i.e., made of hundreds of events, such as in 1994 Paradox Valley, 2006 Basel and 2014 Newberry).
e parameters are estimated using a moving window with constant event number n = 100. Before n is reached,
MLE estimates obtained in retrospect (Table2) are used, as shown in Figure3b. In a prospective case, generic
values should be used, e.g. the median or mean of values taken by the parameters in previous experiments. Since
the post-injection phase is not considered in the sensitivity analysis, a is directly obtained from Eq. (1) so that
=+ −−
+
abmnVVloglog () (6)
fb iii,0
10 10
1
with i the incremental window step. Noteworthy, n ≥ 100 is reached only for high a and/or low completeness Mc.
Depending on the underground a priori knowledge, the injection prole plan, and the safety criterion (Eq.3), one
may estimate what a reasonable Mc would be for parameter updating over time. is may require further seismic
network planning, with the number and spatial arrangement of seismic stations potentially derived from the
simple function
Mdk cdkc(, )()
cc
13
2
=+
, where d is the distance to the kth nearest seismic station and c1, c2 and
c3 empirical parameters42 (although there are additional theoretical limits on event detection that must be taken
into account43).
Post-injection data analysis. While the linear relationship between λ and
V

is well established17–19, the
pure exponential behaviour of the induced seismicity post-injection tail has only been demonstrated for the 2006
Basel case17. Here, we compare three relaxation models: pure power law
tt()λ∝α−
, pure exponential
λ τ∝−tt() exp( /)
and stretched exponential
tt t() exp( (/ ))
1
λ τ∝−
ββ−
by using the Akaike Information
Criterion44,45. We nd that, of the 7 datasets (discarding KTB94a, which tail is likely cut), 5 are best described by
the pure exponential function. e 2 others are best described by a stretched exponential function with stretching
parameter β = 0.9 and 0.7, for PV94 and NB14b, respectively (β = 1 representing the pure exponential). is jus-
ties the use of a pure exponential in Eq. (1) hence limiting θ to a simple three-parameter set.
Model goodness-of-t. Figure1 shows the general agreement between model and data by visual inspec-
tion. Additionally we test Eq. (1) against our datasets using the Kolmogorov-Smirnov (KS) condence bounds46.
We rst convert the dataset
t()
into a transformed dataset
()
∼
Τ

as follows:
ttt
tdt
[, ,, ,] [, ,,,]
(, )
(7)
tin in
i
t
() 1
()
1
0
i
∫λθ











=……→ =Τ…Τ …Τ
Τ=
∼
Τ

e two datasets are equivalent but with the distribution of
∼
Τ()

being the one of a uniform Poisson process with
unit rate. en using the KS condence bounds, we estimate graphically whether the empirical cumulative distri-
bution function (CDF) of

∼
Τ()
,

F()Τ
∼
, deviates signicantly from the CDF of the uniform distribution, FU(Τ).
Here, we use the confidence bounds not to perform a KS statistical test but rather to examine the
physical-engineering evidence against the proposed model. When
Τ
∼
F()
is within the 95–99% KS bounds, we
classify the model as performing well; when
Τ
∼
F()
falls locally outside the 95–99% KS bounds, we classify the
model as performing fairly well; when
F()
Τ
∼
falls extensively outside the 95–99% KS bounds, we classify the
model as performing poorly. Results are shown in Figure4 with the dashed lines representing the two-sided 95%
and 99% condence intervals. e following conclusions can be drawn: e model performs well for the datasets
KTB94b, B06 and G11; fairly well for the datasets KTB94a, PV94, NB12 and NB14b; and poorly for dataset
NB14a. Given the range of dierent datasets, we conclude that the rate model expressed in Eq. (1) describes fairly
well the relationship between injection ow rate and uid-induced seismicity rate.
Time series simulation. We simulate induced seismicity time series with time-dependent ow rates
V

(t)
and θ(t) = [a(t), b(t), τ] by using the thinning method for the injection phase47. For the post-injection phase for
which τ is assumed constant, as no control over post-injection seismicity is possible, the inversion method is used
instead to simulate event occurrence times t48. Magnitudes m are also simulated using the inversion method (for
both injection and post-injection phases with b time-dependent in the rst phase).
Risk-to-magnitude mapping. We translate the safety threshold IR = 10−6 as Pr(I = 9) = 10−5 assuming
that a statistically average “poor” building collapse is reached for a ground intensity I = 9 (i.e., most buildings
of class A in the EMS98 code suer collapse)49 and that once a building collapses, there is a 10% chance of indi-
vidual fatality50. e magnitude msaf is then estimated from an intensity prediction equation (IPE). For induced
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SCIEnTIFIC RePoRTs | 7: 13607 | DOI:10.1038/s41598-017-13585-9
seismicity, we use the IPE derived from the U.S. Geological Survey “Did You Feel It?” rich database51, corrected
for induced seismicity52:
()
Id ccmcmcdcdcmd(6)( 6) loglog 3(8)
hyptecto tectohyp hyptecto hyp12 3
2
4
10
56
10
σ=+ −+ −+ ++ +
with mtecto the earthquake magnitude in the tectonic context,
=+ddh
hyp22
the hypocentral distance in km
(point source hypothesis at depth h = 4 km), c1 = 11.72, c2 = 2.36, c3 = 0.1155, c4 = −0.44, c5 = −0.002044,
c6 = −0.479 (38), σ = 0.4, msaf = mtecto + mcorr and mcorr = 0.82. e correction is based on the observation that
induced earthquakes would seem less severe in average than tectonic ones. We assume that the Modied Mercalli
Intensity (MMI)49 and the USGS Community Internet Intensity (CII)51,52 are equivalent for sake of simplicity. For a
specic project, a fully probabilistic risk approach9 is recommended to derive the parameters Y and msaf of the ATLS
closed-form expressions (Eqs2–3). Here, the safety thresholds shown in Figure2 use Y = 10−5 with
msaf(d = 0 km) = 5.8 and msaf(d = 50 km) = 7.9 (equivalent in the tectonic case to 5.0 and 7.1, respectively). is sim-
plied approach is used to illustrate our ATLS proof-of-concept for a general case with no site-specic conditions. A
detailed risk analysis, which would integrate risk over the full magnitude range9, remains outside the scope of the
present study. e high values of msaf are due to using IR as the safety metric. Some experts may disagree that such
high magnitudes can be reached in the induced seismicity context31, although it is statistically plausible19. e choice
of the safety metric has however no impact on the validity of the proposed method. For example, using the minor
Figure 4. Goodness-of-t of the model represented by Eq. (1). Using the Kolmogorov-Smirnov (KS)
condence bounds, we nd that the model performs well for the datasets KTB94b, B06 and G11; fairly well
for the datasets KTB94a, PV94, NB12 and NB14b; and poorly for dataset NB14a (see the Methods section for
details).
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9
SCIEnTIFIC RePoRTs | 7: 13607 | DOI:10.1038/s41598-017-13585-9
damage threshold Pr(I = 6) = Y2 (i.e., most buildings of class A in the EMS98 code suer negligible to slight damage)
would yield msaf2(d = 0 km) = 4.0. From Eq. (3), the same ATLS mth would be computed for both Pr(I = 9) = 10−5 and
Pr(I = 6) ≈ 10−3.1 assuming e.g. b = 1 in log10(Y2) = log10(Y) + b(msaf − msaf2).
Decision variable denition. e threshold msaf is xed such that Pr(m ≥ msaf) = Y. Assuming that earth-
quakes follow a non-homogeneous Poisson process,
() ()
mm Tm mYPr 1exp ,
(9)
safsaf
≥=−


−Λ ≥


=
where ∫λ
Λ
≥=
≥θ
Tm mtmm dt(, )(
,;
)
saf
T
saf
0
is the mean cumulative number of events and T is the observa-
tion time. For suciently large T (i.e. T >> tshut-in + τ),
Tm mVtVt(, )10[()()]
(10)
saf
abm
shut in shut in
fb saf

τΛ≥=+
−
−−
with V(t) the cumulative injected volume and Λ(T, m ≥ msaf) ≈ Y for Pr(m ≥ msaf) << 1. Eq. (2) is then obtained
by injecting Eq. (10) into Eq. (9) (note that one could replace Y by –ln(1-Y) in Eq. (2) with an impact on the
results only if Y was tending to 1, which is unlikely in safety norms). We then dene the ATLS as the operational
magnitude threshold mth at which the injection is stopped in order to meet the safety target. Note that mth also
provides the completeness magnitude to attain in the region and thus a basis for seismic network monitoring
planning42,53. We thus obtain the following system of equations (for Y << 1):

τ







+≈
=
−−
−−
−Vt Vt Y
Vt
10 [()()]
10 ()1
(11)
a
shut in shut in
abm
shut in
fb bmsaf
fb th
e second equation is always true since the expected number of events with m ≥ mth is one, given the assumption
that injection is stopped for t = tshut-in as soon as m ≥ mth is rst observed. Substituting V(tshut-in) in the rst equa-
tion of this system yields Eq. (3). It is worth noting that omitting the post-injection tail eect with τ = 0 yields the
basic frequency-magnitude distribution threshold mth = log10(Y)/b + msaf.
Data availability. All the data used in this study are publicly available. For more information, please contact
arnaud.mignan@sed.ethz.ch.
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Acknowledgements
We thank three anonymous reviewers for their comments. is work was funded by the Swiss Competence
Center for Energy Research - Supply of Electricity (SCCER SoE) and by the European DESTRESS project, grant
no. 691728.
Author Contributions
A.M. developed the statistical model and closed-form T.L.S. and tested them on real data and simulations; M.B.
made the mapping between risk and magnitude, as well as the K.S. test; S.W. and D.G. initiated the development
and use of an adaptive T.L.S. based on standard safety measures.
Additional Information
Competing Interests: e authors declare that they have no competing interests.
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