## 1 Introduction

[2] Fluid-induced microearthquakes in hydrocarbon or geothermal reservoirs, aftershocks of tectonic earthquakes or seismic emission in rock samples, are examples of seismicity resulting from a seismogenic activation of finite volumes of rocks. Fluid-induced earthquakes of magnitudes 3 to 4 occurred at several Enhanced Geothermal Systems (EGS) like those of Basel, Cooper Basin, The Geysers field, and Soultz [*Giardini*, 2009; *Majer et al.*, 2007; *Häring et al.*, 2008; *Dyer et al.*, 2008; *Baisch et al.*, 2009]. It seems that smaller but still perceptible events can be also observed by hydraulic fracturing of hydrocarbon reservoirs (see http://www.cuadrillaresources.com/news/). Induced seismic hazard becomes a topic of significance in the shale-gas industry (see http://www.energy.senate.gov/public/index.cfm/hearings-and-business-meetings?ID=2c908340-a9bb-40b4-bf7f-8308b272893d). Its understanding is of a considerable importance for mining of deep geothermic energy [*Giardini*, 2009; *Majer et al.*, 2007; *Cornet et al.*, 2007; *Häring et al.*, 2008]. It is of significance for CO_{2} underground storage [see *Zoback and Gorelick*, 2012] and possibly also for other types of geo-technological activity [see *Avouac*, 2012].

[3] Similarly to the tectonic seismicity, statistics of the induced seismicity can be rather well described by the Gutenberg-Richter frequency-magnitude distribution [*Shapiro et al.*, 2007, 2010, 2011; *Shapiro and Dinske*, 2009b; *Dinske and Shapiro*, 2013]. However, large-magnitude events deviate from these statistics [*Shapiro et al.*, 2011]. In this paper we theoretically and numerically analyze the influence of the finiteness of a perturbed volume on the frequency-magnitude statistics of induced events. In contrast to exact specific mechanical models of rupturing a given pressurized fault (see the recent detailed analysis by *Garagash and Germanovich* [2012]), our analysis is a phenomenological one. It considers many faults and attempts to relate the geometry of the stimulated volume to observable statistical features of the induced seismicity using rather general heuristic assumptions. It is possibly applicable to different types of the seismicity triggering physics like a triggering by pore-pressure perturbations or stress perturbations or a triggering by rate-and-state processes modifying the friction. On the other hand, we describe different statistical scenarios of the triggering process. A tendency of real statistics of the induced seismicity to follow one or another scenario may be indicative for the physics of event nucleation.

[4] We start our analysis with a brief overview of a statistical model of induced seismicity, which ignores the fact that rupture surfaces and stimulated volumes are finite. This model adequately describes a general dependence of the seismicity on the volume of the injected fluid. It introduces also some other seismicity-controlling parameters like the seismogenic index. Then we shortly review data-based indications that the large-magnitude events have different statistical features than numerous small events. We observe that large-magnitude events are underrepresented compared with the predictions of the model mentioned above. Thus, we consider different scenarios of geometric relationships between the stimulated volume and potential rupture surfaces. On this basis, we propose lower and upper bounds of the occurrence probability of given-magnitude induced events. These bounds take into account the finiteness of the rupture surfaces and of the stimulated volumes. The bounds are computed from the statistics of arbitrary-size arbitrary-oriented penny-shaped inclusions (representing potential rupture planes) intersecting a finite ellipsoidal stimulated rock volume. A cuboidal stimulated volume is also considered for preferentially oriented rupture surfaces. We show how the finiteness of rupture surfaces and of the stimulated volume influences the frequency-magnitude relation of the induced seismicity. We also show how an estimate of an averaged stress drop can be obtained from the statistics of seismicity. Further we discuss applications of our results to some borehole-injection-based case histories. The observed frequency-magnitude curves seem to follow mainly the lower bound of the occurrence probability of given-magnitude induced events. However, in some case studies there are individual large-magnitude events clearly deviating from the lower-bound statistic. We propose that such events can be interpreted as triggered ones, in contrast to the absolute majority of other events which we call the induced ones. Then we discuss this interpretation.