Fluid injections in geothermic systems usually induce small earthquakes ( -3<M<2). Occasionally, however, earthquakes with larger magnitudes (M ~ 4) occur. We show that under rather general conditions a probability of an event with a magnitude larger than a given one increases proportionally to the injected fluid mass. We find that the process of pressure diffusion in a poroelastic medium with randomly distributed sub-critical cracks obeying a Gutenberg-Richter relation well explains our observations. The magnitude distribution is mainly inherited from the statistics of pre-existing fracture systems. The number of earthquakes greater than a given magnitude also increases with the strength of the injection source and the tectonic activity of the injection site. Our formulation provides a way to estimate expected magnitudes of induced earthquakes. It can be used to avoid significant earthquakes by correspondingly planning fluid injections.

An important characteristics of seismicity is its magnitude distribution. Fluid injections in rocks aimed to create Enhanced Geothermic Systems (EGS) can sometimes produce significant seismic events (see e.g., Majer et al. (2007)). To our knowledge, this is less frequently the case for stimulations of hydrocarbon reservoirs. Here we formulate a theoretical model describing distribution of magnitudes of induced seismicity with time. We show that this model is in a well agreement with real data from geothermic and hydrocarbon reservoirs. Shapiro et al. (2007) described magnitude distribution in the case of linear fluid-rock interactions (i.e., linear diffusion of pore pressure perturbations caused by fluid injections into rocks) and constant injection rates. Here, we significantly generalize this approach. We extend it to a rather general type of injection sources. We include a possibility that the fluid-rock interaction can be strongly non-linear in the sense of a strong impact of the fluid injection onto rock permeability. Such situations like a hydraulic fracturing or a shear-caused dilatational permeability enhancement are taken into account.


We consider a pressure source in an infinite, homogeneous, permeable, porous continuum. Due to a fluid injection and the consequent process of pressure relaxation, the pore pressure p will change throughout the pore space. We assume that a random set of preexisting cracks (defects) is statistically homogeneously distributed in the medium and is characterised by the volume concentration N. For simplicity we assume that the cracks do not mutually interact. Each of these cracks is characterized by an individual critical value C of the pore pressure necessary in accordance with the Coulomb failure criterion for occurrence of an earthquake along such a defect. This critical pressure C is randomly distributed on a set of pre-existing cracks. If C is high we speak about a stable preexisting fracture. If C is low we mean a fracture close to its failure. Statistical properties of C are assumed to be independent of spatial locations (i.e., C(r) is a statistically homogeneous random field). If at a given point r of the medium (with a pre-existing crack there) pore pressure p(t, r) increases with time, and at time t0 it becomes equal to C(r) then this point will be considered as a hypocentre of an earthquake occurring at this location at time t0.

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