# Proportional hazards and finite size effects in a conceptual model of fluid-induced microseismicity

Jordi Baro,
Cole Lord-May,
Jörn Davidsen,
David Eaton

Natural and anthropogenic processes involving the injection or migration of fluids within rock formations are well known driving mechanisms causing earthquakes, either induced or triggered. A variety of field observations has led to the formulation of three different modeling paradigms for the estimation of the seismic hazards as function of injected volume ($V$) [1], useful to develop risk mitigation strategies [2]. All of them are based on proportional hazard models but imposing different constraints caused by finite size effects:

A) events are small and their number is proportional to injected volume: ($N(m)\propto V$) [3,4];

B) events are bounded in size and magnitude ($m$) by the extension of the stimulated area and other physical constrains [5];

C) event magnitudes are statistically indistinguishable from tectonic earthquakes [6].

Based on the simulation results of a simplified conceptual model accounting for the non-homogeneous pore-pressure stimulation [8] caused by fluid injection in a pre-stressed region with stick-slip mechanics [7], we provide a fundamental explanation on how all three paradigms can naturally coexist. The loading history and heterogeneity of the host medium determine which of the three paradigms dominates. In non-tectonic settings two populations of events triggered at different pore-pressure levels with different Gutenberg-Richter $b$-values are superposed. Stress-levels an d $V$ determine their proportions. In active tectonic settings, fluid injection triggers tectonic earthquakes.

[1] D. W. Eaton, N. Igonin (2020) The Leading Edge 37:2, 135-140 10.1190/tle37020135.1
[2] T. Ader, et al. (2019), J. Seismol. 10.1007/s10950-019-09853-y.
[3] A. McGarr (1976), J. of Geophs. Res. 81, 1487 10.1029/JB081i008p01487.
[4] C. Dinske, S. Shapiro (2013), J. Seismol. 17: 13  10.1007/s10950-012-9292-9.
[5] S. Shapiro, O. Krüger C. Dinske, (2013) J. of Geophs. Res. Solid Earth 118, 3557 10.1002/jgrb.50264.
[6] N. van der Elst et al. (2016), J. of Geophs. Res. Solid Earth 121, 4575 10.1002/2016JB012818.
[7] Y. Ben-Zion, J. Rice (1993), J. of Geophs. Res. Solid Earth 98, 14109 10.1029/93JB01096.
[8] J. Norris, D. Turcotte, J. Rundle (2014), Physical Review E 89, 022119 10.1103/PhysRevE.89.022119.

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