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.

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

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 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

Seismicity spatio-temporal behaviour described by the N-C PAST static
stress model (tectonic case; Mignan, 2012):

In the tectonic case, static stress changes are underloading due to
hypothetical precursory silent slip on the fault at

In the case of an EGS stimulation, the stress source is the fluid injected at
depth with overpressure

In the EGS case,

The induced seismicity rate

Figure 2 shows the flow rate

I evaluate

2006 Basel EGS stimulation experiment data with activation and
quiescence envelope fits:

Induced seismicity production time series, observed and predicted:

The density of events above

Figure 3 shows the 6 h rate of induced seismicity

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

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

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

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 (

The main assumption of the N-C PAST is to consider three unique seismicity
regimes (quiescence, background, and activation) defined by the event
productions

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