Modeling Time Dependent Earthquake Magnitude Distributions Associated with Injection-Induced Seismicity

Understanding and predicting earthquake magnitudes from injection-induced seismicity is critically important for estimating hazard due to injection operations. A particular problem has been that the largest event often occurs post shut-in. A rigorous analysis would require modeling all stages of earthquake nucleation, propagation, and arrest, and not just initiation. We present a simple conceptual model for predicting the distribution of earthquake magnitudes during and following injection, building on the analysis of Segall & Lu (2015). The analysis requires several assumptions: (1) the distribution of source dimensions follows a Gutenberg-Richter distribution; (2) in environments where the background ratio of shear to effective normal stress is low, the size of induced events is limited by the volume perturbed by injection (e.g., Shapiro et al., 2013; McGarr, 2014), and (3) the perturbed volume can be approximated by diffusion in a homogeneous medium. Evidence for the second assumption comes from numerical studies that indicate the background ratio of shear to normal stress controls how far an earthquake rupture, once initiated, can grow (Dunham et al., 2011; Schmitt et al., submitted). We derive analytical expressions that give the rate of events of a given magnitude as the product of three terms: the time-dependent rate of nucleations, the probability of nucleating on a source of given size (from the Gutenberg-Richter distribution), and a time-dependent geometrical factor. We verify our results using simulations and demonstrate characteristics observed in real induced sequences, such as time-dependent b-values and the occurrence of the largest event post injection. We compare results to Segall & Lu (2015) as well as example datasets. Future work includes using 2D numerical simulations to test our results and assumptions; in particular, investigating how background shear stress and fault roughness control rupture extent.