Stimulating rupture surfaces in a finite rock volume

Pore fluids in rocks and pore pressure perturbations can trigger earthquakes. Sometimes fluid injections into boreholes are able to induce potentially damaging seismic events. For instance, this was the case by stimulations at such Enhanced Geothermal Systems like the ones at Basel, in Cooper Basin, at The Geysers field and at Soultz. 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. Such a finiteness can influence frequency-magnitude statistics of the seismicity. Previously we have observed that fluid-induced large-magnitude events at geothermal and hydrocarbon reservoirs are frequently underrepresented in comparison with the Gutenberg-Richter statistics. This is an indication that the events are much more probable on rupture surfaces contained nearly completely within the stimulated volume. Here we theoretically analyse the influence of the finiteness of a perturbed volume on the frequency-magnitude statistics of induced events. Our analysis is a phenomenological one. It is possibly applicable to different types of the seismicity triggering like a triggering by pore-pressure perturbations or a triggering by rate-and-state processes. We approximate a stimulated volume by an ellipsoid or cuboid, and derive the magnitude statistics of induced events from the statistics of randomly orientated thin flat discs of different sizes, representing the rupture surfaces. We consider different possible scenarios of event triggering: rupture surfaces located completely within the stimulated volume and rupture surfaces which are intersecting with the stimulated volume. We derive lower and upper bounds of the probability to induce a given-magnitude event. The bounds depend on the characteristic scales of the stimulated volume. The minimum principal axis is the most influential geometric parameter. We compare our analytical results with data on seismicity induced by fluid injections in boreholes. Fitting the bounds to the frequency-magnitude distribution can provide an estimate of a largest expected magnitude and a characteristic stress drop, in addition to improved estimates of the Gutenberg-Richter a- and b- parameters.