Constitutive Law for Earthquake Production Based on Rate‐and‐State Friction: Theory and Application of Interacting Sources

Abstract

The constitutive relationship of Dieterich (1994, https://doi.org/10.1029/93JB02581) for earthquake production has been applied widely to earthquake triggering. The Dieterich theory achieves mathematical simplicity and thus lends itself to practical applications through a number of simplifying assumptions. Perhaps the most suspect and criticized assumption is that seismic sources do not interact. Here I provide an extension of the constitutive framework, using the modified formulation of Heimisson and Segall (2018, https://doi.org/10.1029/2018JB015656) which accounts for source interactions. Moreover, I explore ways to account for earthquake magnitudes and highlight potential directions for future research. I identify conditions under which the constitutive relationship with interactions reduces to the noninteracting Dieterich theory, revealing that an interacting population may be modeled as an equivalent noninteracting population. This result may partly explain the success of the Dieterich theory in spite of assuming no interactions. These special conditions reveal that if the magnitude distribution (e.g., the b value) is altered by a stress perturbation, then the Dieterich model may either underpredict or overpredict the seismicity rate. Furthermore, the interaction model grants insight into the physical interpretation of the background seismicity rate, which is frequently considered to be a declustered rate. However, the results suggest that interactions do not change the total cumulative number of events over a long time span (t ≫ ta). Finally, I apply the interaction theory to fault slip models, aftershock diffusion, and secondary triggering as illustrative examples.

Physically Consistent Modeling of Dike‐Induced Deformation and Seismicity: Application to the 2014 Bárðarbunga Dike, Iceland

“…This indicates that a population with interactions can be approximated as a population without interactions with the same long‐term average background seismicity rate. In addition, Heimisson () showed that interaction between populations in a spatially heterogeneous stress field do not change the absolute number of events on a regional scale for times . This suggests that interactions do not change the absolute number of events, although they may somewhat change their temporal and spatial distribution.…”

Clustering characteristics of gas-extraction induced seismicity in the Groningen gas field

“…In Groningen, apart from the absence of any (known) tectonic activity, no seismicity was observed for the first 20-25 yr of gas production. This absence of seismicity implies that no fault segments within the Groningen gas field have near-threshold initial stresses at the onset of depletion (Muntendam-Bos & de Waal 2013;Wees et al 2014;Bourne & Oates 2017;Dempsey & Suckale 2017;Candela et al 2018) or the tectonic loading rate is very low, resulting in a very low background seismicity rate (Dieterich 1994;Heimisson 2019). In terms of changes in effective stress, the Mohr-Coulomb circle is far from the failure line, but slowly grows with ongoing pressure depletion (Fig.…”

Physically consistent modeling of dike induced deformation and seismicity: Application to the 2014 Bárðarbunga dike, Iceland

“…This indicates that a population with interactions can be approximated as a population without interactions with the same long term average background seismicity rate. In addition, Heimisson [2019] showed that interaction between populations in a spatially heterogeneous stress field, do not change the absolute number of events on a regional scale for times t t a . This suggests that interactions do not change the absolute number of events, although they may somewhat change their temporal and spatial distribution.…”

“…However, the population of seismic sources (see Heimisson [2019] for precise definition) may not have been sufficiently stressed prior to the intrusion to produce earthquakes at a constant rate. Indeed, Figure 5 suggests that in most places the background shear to normal stress ratio was fairly small.…”