NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-23-107-2016Static behaviour of induced seismicityMignanArnaudarnaud.mignan@sed.ethz.chhttps://orcid.org/0000-0002-2167-7534Institute of Geophysics, ETH Zurich, Zurich, SwitzerlandArnaud Mignan (arnaud.mignan@sed.ethz.ch)29April20162321071136October201510December201515February201613April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/23/107/2016/npg-23-107-2016.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/23/107/2016/npg-23-107-2016.pdf
The standard paradigm to describe seismicity induced by fluid injection is to
apply non-linear diffusion dynamics in a poroelastic medium. I show that the
spatio-temporal behaviour and rate evolution of induced seismicity can,
instead, be expressed by geometric operations on a static stress field
produced by volume change at depth. I obtain laws similar in form to the ones
derived from poroelasticity while requiring a lower description length.
Although fluid flow is known to occur in the ground, it is not pertinent to
the geometrical description of the spatio-temporal patterns of induced
seismicity. The proposed model is equivalent to the static stress model for
tectonic foreshocks generated by the Non-Critical Precursory Accelerating
Seismicity Theory. This study hence verifies the explanatory power of this
theory outside of its original scope and provides an alternative physical
approach to poroelasticity for the modelling of induced seismicity. The
applicability of the proposed geometrical approach is illustrated for the
case of the 2006, Basel enhanced geothermal system stimulation experiment.
Applicability to more problematic cases where the stress field may be
spatially heterogeneous is also discussed.
Introduction
Induced seismicity is a growing concern for the energy
industry relying on fluid injection in the deep parts of the Earth's crust
(Ellsworth, 2013; Mignan et al., 2015). At the same time, fluid injection
sites provide natural laboratories to study the impact of increased fluid
pressure on earthquake generation (Majer et al., 2007). Induced seismicity is
characterised by two empirical laws, namely (i) a linear relationship between
the fluid mass m(t) injected up to time t and the cumulative number of
induced earthquakes N(t) and (ii) a parabolic induced seismicity spatial
envelope radius r∝m(t)n, with n being a positive integer
(Shapiro and Dinske, 2009). These two simple descriptive laws can be derived
from the differential equations of poroelasticity (Biot, 1941) under various
assumptions (Shapiro and Dinske, 2009). In general however, the full
description of the process requires complex numeric modelling coupling fluid
flow, heat transport, and geomechanics (Rutqvist, 2011). These models,
numerically cumbersome, can become intractable because of the sheer number of
parameters (Miller, 2015). Attempts to additionally correct for the known
discrepancies between Biot's theory and rock experiments have led to a large
variety of model assumptions (Berryman and Wang, 2001), indicating that
poroelasticity results are ambiguous.
I will demonstrate that a static stress model can explain the two empirical
laws of induced seismicity without requiring any concept of poroelasticity.
The proposed theoretical framework hence avoids the aforementioned
shortcomings by suggesting an origin of induced seismicity that does not
involve fluid flow in a porous medium (although fluid flow indeed occurs).
Historically, a similar static stress model was proposed for the tectonic
regime under the Non-Critical Precursory Accelerating Seismicity Theory (N-C
PAST) (Mignan et al., 2007; Mignan, 2008, 2012). Its application to induced
seismicity data will allow a more fundamental investigation of the
relationship between static stress and earthquake generation. To test the
model, I will use data from the 2006 Basel enhanced geothermal system (EGS)
stimulation experiment, including the flow rate of injected fluids (Häring
et al., 2008) and the relocated catalogue of induced seismicity (Kraft and
Deichmann, 2014).
The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
The N-C PAST has been proposed to explain the precursory seismicity patterns
observed before large earthquakes from geometric operations in the
spatio-temporal stress field generated by constant tectonic stress
accumulation (Mignan et al., 2007; Mignan, 2008, 2012). In particular, it
provides a physical algebraic expression of temporal power laws without
requiring local interactions between the elements of the system (Sammis and
Sornette, 2002; Mignan, 2011). Therefore earthquakes are considered passive
(static) tracers of the stress accumulation process, in contrast with active
earthquake cascading in a critical process (hence the term “non-critical”).
The concept of self-organised criticality (Bak and Tang, 1989) is seldom used
to explain induced seismicity (Grasso and Sornette, 1998). Since there is no
equivalent of a mainshock in induced seismicity, the
criticality-versus-non-criticality debate has limited meaning in that case. However, the
underlying process of static stress changes considered in the N-C PAST can be
tested against the observed spatio-temporal behaviour of induced seismicity.
The N-C PAST postulates that earthquake activity can be categorised into three
regimes – background, quiescence, and activation – depending on the
spatio-temporal stress field σ(r,t):
σr,t=σ0∗,t<t0hnr2+h2n2σ0+τ˙t-t0+σ0∗,t0≤t<tf
defined from the boundary conditions σ(r+∞,t)=σ0∗ and σ(r=0,t)=σ0+τ˙t+σ0∗, with h being the depth of the fault segment base, r the
distance along the stress field gradient from the fault's surface projection,
σ0< 0 the stress drop associated with a hypothetical silent slip
occurring at t0 at the base of the fault, τ˙ the tectonic
stress rate on the fault, σ0∗ the crustal background stress,
n=3 the spatial diffusion exponent for static stress, and tf
the mainshock occurrence time (Mignan et al., 2007) (Fig. 1a). Background,
quiescence, and activation regimes are strictly defined by the three event
spatio-temporal densities δb0, δbm, and
δbp for |σ|≤σ0∗±Δσ∗, σ<σ0∗-Δσ∗, and σ>σ0∗+Δσ∗, respectively, with the boundary
layer being ±Δσ∗ the background stress amplitude range
(so-called N-C PAST postulate). By definition, δbm <δb0 <δbp, with each seismicity regime
so far being assumed isotropic and homogeneous in space (i.e. the role of the fault
network is neglected). Correlation between earthquake productivity and static
stress changes is well established (King, 2007). The distinction of three
unique seismicity regimes with constant event density, the main assumption of
the N-C PAST, is discussed later on.
Seismicity spatio-temporal behaviour described by the N-C PAST static
stress model (tectonic case; Mignan, 2012): (a) spatio-temporal
evolution of the stress field σ(r,t) generated by constant stress
accumulation τ˙ on a fault located at r=0 (Eq. 1). Background,
quiescence, and activation seismicity regimes are described by densities of
events δb0, δbm, and δbp for
|σ|≤σ0∗±Δσ∗, σ<σ0∗-Δσ∗, and σ>σ0∗+Δσ∗,
respectively; (b) temporal evolution of quiescence and activation
envelopes r∗(t), with σr∗=σ0∗±Δσ∗ (Eq. 2); (c) rate of events
μ(t) in a disc of constant radius maxr∗ (Eq. 3);
(d) cumulative number of events N(t) (Eq. 4) of power law form
(Eq. 5), with t0= 0, tmid= 1, tf= 2, h= 1,
τ˙= 0.1, σ0∗= 0, Δσ∗=10-2, δbm= 0.001, δb0= 0.1,
δbp= 1, n=3, k=π, d=2, Δt= 0.01.
In the tectonic case, static stress changes are underloading due to
hypothetical precursory silent slip on the fault at t0 followed by
overloading due to hypothetical asperities delaying rupture on the fault
after tp∗ (Mignan, 2012). The three seismicity regimes are
then defined as solid spatio-temporal objects with envelopes
rQ∗t0≤t<tm∗=hτ˙tm∗-tΔσ∗+12/n-11/2rA∗tp∗<t<tf=hτ˙t-tp∗Δσ∗+12/n-11/2
by applying to Eq. (1) the boundary conditions σrQ∗,t=σ0,tm∗=σ0∗-Δσ∗ and σrA∗,t=σ0,tp∗=σ0∗+Δσ∗, respectively. The
parameters tm∗=tmid-Δσ∗/τ˙ and tp∗=tmid+Δσ∗/τ˙ represent the times of quiescence disappearance
and of activation appearance, respectively, with σ0,tmid=σ0∗. The background seismicity regime is
defined by subtracting the quiescence and activation envelopes
rA∗(t) and rQ∗(t) from a larger constant
envelope rmax≥maxrA∗,rQ∗ to avoid truncating the quiescence and
activation solids (Fig. 1b). While trivial along r, concepts of
geometric modelling may be required to represent these seismicity solids in
three-dimensional Euclidian space (Gallier, 1999) in which the vector
r may change direction in space (Mignan, 2008, 2011). The
non-stationary background seismicity rate μ(t) is then defined in the
volume of maximum extent rmax by
μt=δb0krmaxd,t<t0δb0krmaxd-rQ∗(t)d+δbmkrQ∗(t)d,t0≤t<tm∗δb0krmaxd,tm∗≤t≤tp∗δb0krmaxd-rA∗(t)d+δbpkrA∗(t)d,tp∗<t<tf
with k being a geometric parameter and d the spatial dimension. For the
tectonic case in which rmax≫h, the volume is assumed to be a
cylinder with k=π, d=2, and δ being the density of epicentres in
space (Fig. 1c). It should be noted that taking rmax very large
relative to maxrA∗,rQ∗
tends to mask the non-stationary seismicity pattern to be investigated. As a
consequence it is preferable in practice to use rmax=maxrA∗,rQ∗. Finally, the
cumulative number of events N(t) is defined as
Nt=∫0tfμtdt,
which represents a power law time-to-failure equation of the form
Nt∝t+tdn+1,
with the first term representing the linear background seismicity and the second
term the quiescence or activation power law behaviour observed prior to some
large mainshocks (Fig. 1d) (see the review by Sammis and Sornette (2002) for
different physical processes yielding a temporal power law).
Application of the N-C PAST static stress model to induced
seismicity
In the case of an EGS stimulation, the stress source is the fluid injected at
depth with overpressure
Pt,r=0=KΔVt,ΔtV0,
where K is the bulk modulus, ΔV the volume change per time unit, and
V0 the infinitesimal volume subjected to pressure effect per time unit at
the borehole located at r=0. The injected volume V(t) is determined
from the flow rate profile Q(t), as
Vt=∫t0tQtdt,
with t0 being the starting time of the injection. The volume change rate is then
defined as
ΔVt,Δt=Vt-Vt-ΔtΔt,
with Δt being a time increment.
In the EGS case, r≅h, with h being the borehole depth and induced
seismicity defined as hypocentres. The spatio-temporal stress field
σr,t becomes
σr,t=σ0∗,t<t0r0nr+r0nPt,r=0+σ0∗,t≥t0
with r being the distance along the stress field gradient from the borehole; n=3 the spatial diffusion exponent for static stress; and r0→0 the
infinitesimal radius of volume V0=kr0d/t0, where t0=1 is the time
unit. The parameter r0 is incidental and disappears in the induced
seismicity case (see below). Activation represents the case when fluids are
injected and quiescence when fluids are ejected (bleed-off), or, in terms of
stress field variations, when the pressure change by fluid injection is
positive or negative, respectively. It follows that
rA∗t|ΔV≥0=r0n-dkKt0Δσ∗ΔV(t)1/n-r0rQ∗t|ΔV<0=-r0n-dkKt0Δσ∗ΔV(t)1/n-r0,
which suggests that the spatio-temporal shape of the induced seismicity
envelope depends on the nth root of the flow rate profile Q(t) (with n=3 in the static stress case). This parabolic relationship is similar to the
generalised form r(t)∝m(t)1/d derived from non-linear
poroelasticity in a heterogeneous medium where m is the cumulative mass of
injected fluid and d the spatial dimension (Shapiro and Dinske, 2009). The
main difference between the two physical approaches is in the underlying
stress field, which is here static and in poroelasticity dynamic and related
to the displacement gradient of the fluid mass (Rudnicki, 1986). As a side
note, it is trivial to derive Eq. (10) from Eq. (9), while numerous
assumptions are necessary to obtain the parabolic form m(t)1/d in
non-linear poroelasticity (Shapiro and Dinske, 2009).
The induced seismicity rate μ(t) is then defined by Eq. (3) but with
r∗ from Eq. (10), k=4π/3, and d=3, assuming a spherical
spatial volume (i.e. isotropic stress field). For the activation phase (i.e.
stimulation period), it follows that
Nt∝ΔV(t)dn+1
or
Nt∝V(t)dn.
The induced seismicity case d=n=3 confirms the linear relationship
between cumulative injected volume and cumulative number of induced
earthquakes N(t)∝V(t) previously derived from poroelasticity (e.g.
Shapiro and Dinske, 2009). In contrast with poroelasticity, this second law
is a direct consequence of the first. The d=n condition also yields the
simplified form of Eq. (10):
rA∗t|ΔV≥0≈34πKt0Δσ∗ΔV(t)1/3rQ∗t|ΔV<0≈-34πKt0Δσ∗ΔV(t)1/3,
where the one free parameter is the normalised background stress amplitude
range Δσ∗^=Δσ∗/Kt0.
Application to the 2006 Basel EGS induced seismicity sequence
Figure 2 shows the flow rate Q(t) of injected fluids during the 2006 Basel
EGS stimulation experiment (Häring et al., 2008) and the spatio-temporal
distribution of relocated induced seismicity (Kraft and Deichmann, 2014)
above completeness magnitude Mc=0.8. The injection started at
18:00 LT on 2 December 2006 (t0) and stopped at 11:33 LT on 8 December
2006 (t1), after which the well was bled off (ΔV< 0) (Fig. 2a).
The N-C PAST thus predicts an activation envelope rA∗ for
t0≤t<t1 and a quiescence envelope rQ∗ for t≥t1 (Eq. 13). The activation and quiescence envelopes are fitted to the
Basel data using Δσ∗^∈[10-3,10-1]day-1 (light curves) and Δt=1/4 day. The
results are shown in Fig. 2b. The value Δσ∗^= 0.007 day-1 (dark curves) provides the best fit to the data,
defined from the best score S=(wA+wQ)/2, with
wA and wQ being the ratio of events of distance r≤rA∗ and r≥rQ∗ in the injection and
bleeding-off phases, respectively. Figure 2c shows S as a function of
Δσ∗^ for Δt=1/12,1/8,1/4 day, which indicates that the results remain stable for lower time
increments.
I evaluate δb0=10-10eventm-3day-1 by
counting all earthquakes declared in the national Swiss catalogue (ECOS-09,
http://hitseddb.ethz.ch:8080/ecos09/) and located within 10 km
of the borehole of coordinates (7.594∘ E; 47.586∘ N) and
depth 4.36 km. This means that ∼ 1 tectonic earthquake is
expected on average in the space–time window considered. Due to the low
tectonic activity in the area, I approximate δb0=δbm= 0 eventm-3day-1 (i.e. total
quiescence). The theory shows a good agreement with the observations, with
97 % of the seismicity below rA∗ during the injection
phase (red points in Fig. 2b) and 98 % of the seismicity above
rQ∗ during the bleeding-off phase (orange to yellow
points).
2006 Basel EGS stimulation experiment data with activation and
quiescence envelope fits: (a) flow rate Q(t) (digitised from
Häring et al., 2008); (b) spatio-temporal distribution of
relocated induced seismicity (Kraft and Deichmann, 2014) with r the
distance from the borehole. The activation and quiescence envelopes
rA∗(t) and rQ∗(t) are defined from
Eq. (13) with parameters Δσ∗^= 0.007 day-1 (dark curves) and Δt=1/4 day. The
light curves represent the range Δσ∗^∈[10-3,10-1]day-1 in 0.1 increments on the log10 scale. Points
represent the induced earthquakes; which colour indicates how they are
declared. (c) Score S=(wA+wQ)/2, with
wA and wQ being the ratio of events of distance r≤rA∗ and r≥rQ∗ in the injection and
bleeding-off phases, respectively. The vertical line represents
Δσ∗^= 0.007 day-1.
Induced seismicity production time series, observed and predicted:
(a) histogram of the observed 6 h induced seismicity rate
μ(t) with fit based on Eq. (15) with MLE parameters δbp=4.68×10-7eventm-3day-1 (production parameter)
and τ= 1.18 day (diffusion parameter); (b) cumulative number
of induced earthquakes N(t) with fit based on Eq. (4) with μ(t) of
Eq. (15).
The density of events above rQ∗ is however not δb0 but an equation of the form
δbt≥t1=δbpexp-t-t1τ,
which represents the temporal diffusion of induced seismicity; τ is the
average time constant (e.g. Mignan, 2015). Diffusion from density
δbp to δb0 was originally not considered in the
N-C PAST as any potential diffusion after an activated foreshock sequence
would be shadowed by the effects of the subsequent mainshock. Here however,
diffusion dominates in the post-injection phase. Equation (14)
represents a relaxation process from the overloading state to the background
state. The results here suggest that only the events declared as background
(grey points) and quiescence events (blue points) are outliers. The observed
variations in r below rA∗ and above rQ∗
are not explained by the model, which only predicts the behaviour of the
activation and quiescence fronts. The second-order variations may be due to
anisotropic effects and for t>tmaxrA∗ to additional spatial
diffusion effects.
Figure 3 shows the 6 h rate of induced seismicity μ(t) and the
cumulative number of induced events N(t), observed and predicted. With
δb0=δbm=0 and taking into account induced
seismicity temporal diffusion, the rate of induced seismicity becomes
μt=max4π3δbp⋅Δt⋅r∗(t)3,4π3δbp⋅Δt⋅r∗t-St3exp-t-Stτ,
where δbp=4.68×10-7eventm-3day-1
(production parameter) and τ= 1.18 day (diffusion parameter) are
obtained by maximum-likelihood estimation (MLE), St=Δt,…,iΔt,…, and
r∗t=0,t<t0rA∗(t),t0≤t<t10,t≥t1.
Equation (15) implies that induced seismicity is fully explained by
overloading, in agreement with the observation of no causal relationships
between events in the Basel sequence (Langenbruch et al., 2011). The
predicted rate (Eq. 15) and predicted cumulative number of events (Eq. 4) fit
the data well, as shown in Fig. 3a and b, respectively. The role of
temporal diffusion is observed after t1-Δt and is the only
contributor to induced seismicity after t1. Of three functional forms
tested to describe diffusion (exponential, stretched exponential, and power
law), the exponential (Eq. 14) was verified to be the best model for the
Basel case (following the formalism and tests proposed by Clauset et
al. (2009); see also Mignan (2015, 2016) for the tectonic aftershock case).
Discussion
The two descriptive laws of induced seismicity (one: linear relationship
between fluid volume injected and cumulative number of events; two:
parabolic spatial envelope) had been previously obtained by considering the
differential equations of poroelasticity (Biot, 1941; Rudnicki, 1986) under a
number of assumptions (Shapiro and Dinske, 2009). The algebraic expressions
derived in the present study from geometric operations on a static stress
field reflect a lower description length of the physical process (Kolmogorov,
1965) since all of Biot's theory is bypassed although similar characteristics
of induced seismicity are modelled at the end (Fig. 4). Although the commonly
used parabolic expression rt=4πDt, with D being the
hydraulic diffusivity and t the time since the injection start (Shapiro et
al., 1997), is relatively simple to derived from linear poroelasticity, it
generally badly describes the early stage of the injection. This led to the
addition of an arbitrary non-zero starting time t0 in previous works
(e.g. Shapiro et al., 2006), including the Basel case (Shapiro and Dinske,
2009), and finally to the consideration of non-linear poroelasticity (ibid.).
The proposal in the present article of a more parsimonious and transparent
approach obviously does not mean that it is superior to poroelasticity. It
should simply be seen as a new alternative to induced seismicity modelling
that is worth exploring in more detail.
Description length defined as the count of physical steps required
to describe induced seismicity, in poroelasticity and in the newly proposed
geometrical approach. In the latter, Biot's theory is entirely bypassed.
The simplicity of the geometrical approach might a priori only appear
applicable to homogeneous cases, such as the 2006 Basel EGS stimulation
example. In fact, the approach could be applied to more problematic data sets
that involve anisotropy and other heterogeneities. The most common example of
anisotropy is the case of induced seismicity being spatially guided by a
fault structure, such as during the 2004–2005 German Continental Deep Drilling Programme (KTB)
injection (Shapiro et al., 2006) or the 2013 St Gallen, Switzerland
stimulation (Edwards et al., 2015). Figure 5 illustrates how such
heterogeneity can be implemented in the geometrical approach, by adding the
historical static stress field that is associated with an active tectonic
fault. It should be noted that this idea was first suggested in Mignan (2011)
to explain the observed variability in tectonic precursory seismicity
patterns. In the example of Fig. 5, a fault is located between 1.5<r<3.0
away from the borehole (r= 0). If a mainshock occurred on that fault in
the distant past, it would have created an overloading field (i.e. σ>σ0∗+Δσ∗). Over time, this static
stress field would have been “planed” to the threshold σ0∗+Δσ∗ by temporal diffusion (represented by aftershocks, i.e.
Eq. 14). This would yield a permanent “ghost” of that historical static
stress field. It follows that during fluid injection there would then be two
clusters of induced seismicity, one spherical, centred on the borehole, and a
second, elongated, following the fault structure (Fig. 5).
Conclusions
I have demonstrated that the two principal induced seismicity
descriptive laws can be explained from geometric operations in a static
stress field without requiring any concept derived from poroelasticity. I
have shown that the controlling parameter is then the normalised background
stress amplitude range Δσ∗^, which questions the
usefulness of permeability and diffusivity parameters in induced seismicity
analyses and might explain why these parameters remain elusive (Miller,
2015). In that view, permeability could depend on the “external loading
configuration” instead of on the material itself, as recently proposed in
the case of the static friction coefficient (Ben-David and Fineberg, 2011).
Testing of the model on other induced seismicity sequences will determine if
Δσ∗^ is itself universal, region-specific, or
related to the static stress memory of the crust, hence whether or not
Δσ∗^ depends on the tectonic loading
configuration at EGS natural laboratory sites. Similar questions apply to the
earthquake production parameter δbp and if the two parameters
are independent or correlated.
Sketch on how anisotropy and other types of heterogeneities can be
implemented in the geometrical approach by adding a historical tectonic
static stress field (ad hoc parameter values used for sake of simplicity).
Here a past overloading field (σ>σ0∗+Δσ∗) on a nearby fault would have been “planed” to the threshold
σ0∗+Δσ∗ by temporal diffusion (Eq. 14),
leaving only a “ghost” of that historical static stress field (for the
homogeneous case, see Fig. 1a).
The main assumption of the N-C PAST is to consider three unique seismicity
regimes (quiescence, background, and activation) defined by the event
productions δbm<δb0<δbp. There are
two possible physical alternatives to justify this choice: (1) it represents
the fundamental behaviour of the Earth's crust, which would hence act as a
capacitor, with strain energy storage and δbp analogues to
electrical energy storage and capacitance, respectively (a parallel between
tectonic aftershocks and a discharging Leyden jar is for instance made in
Mignan, 2016); (2) the proposed step function is a simplification of the true
stress-production profile, which remains unknown and is so far best
characterised by three regimes (e.g. King, 2007). Both alternatives allow
defining spatio-temporal solids over which geometric operations yield
algebraic expressions of the induced seismicity behaviour.
Acknowledgements
I thank two anonymous reviewers for their comments. The work leading to this
article was funded by the Swiss Competence Center for Energy Research –
Supply of Electricity (SCCER-SoE).
Edited by: R. Gloaguen
Reviewed by: three anonymous referees
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