Modelling fluid-induced seismicity rates associated with fluid injections: examples related to fracture stimulations in geothermal areas

In this paper, we present a model for describing relationships between fluid-induced seismicity and operational parameters of fluid injections. Considering seismic sequences occurring during sustained fluid injections, we present a novel covariate approach in which a probability distribution is defined as a basic template function for modelling the interevent times (i.e. the time intervals between consecutive events), and the possible dependencies on operational parameters are modelled writing the parameters of the probabilistic model in terms of deterministic functions of explanatory covariates that are selected from the operational data. The implemented model is tested using data from two cases of reservoir stimulations in geothermal systems (The Geysers, US and Cooper Basin, Australia). We have found that a template exponential distribution of interevent times, with a linear function relating the logarithm of the distribution's mu parameter and the logarithm of the injection rate, is the model that better describes the observations in all the analysed cases. This result suggests that the mu parameter and the injection rate have a power-law relationship. The value of the power-law exponent, alpha(1), is an indicator of the relative change in the seismicity rate associated with a relative change in the injection rate, and it results particularly important for understanding the behaviour of seismicity at high injection rates. alpha(1) = -1 indicates the special case of a linear relationship between seismicity rate and injection rate (which usually is the model assumed in the literature); conversely, alpha(1) > -1 (alpha(1) < -1) indicates that the relative change in the seismicity rate associated with a relative change in the injection rate is lower (higher) than the relative change expected by assuming a linear relationship between these two parameters. Regarding the cases analysed in this paper, the exponent of the power law varies between -1.04 and -0.78 in The Geysers, and between -1.22 and -0.73 in Cooper Basin. While no clear temporal trends are observable in the alpha(1) values obtained for The Geysers, in Cooper Basin alpha(1) increased as the stimulation tests proceeded (from an initial value of alpha(1) = -1.22 calculated for the first fracture initiation test to alpha(1) = -0.73 associated with the third fracture initiation test). This behaviour can be a consequence of the so-called Kaiser effect indicating that, to trigger events, the pore pressure during new injections must exceed the values already reached in previous injection operations. Finally, we also studied the gradual decline of seismicity rates in the post-injection phases in Cooper Basin using the modified Omori law. We explored this data set for looking, in particular, if there exists a pattern between the parameters controlling the post-injection seismicity decay rate and the characteristics of the precedent fluid injection.