High-resolution computer simulations of earthquake sequences in three or even two dimensions pose great demands on time and energy, making lower-cost simplifications a competitive alternative. We systematically study the advantages and limitations of simplifications that eliminate spatial dimensions, from 3D down to 0/1D in quasi-dynamic earthquake sequence models. We demonstrate that, when 2D or 3D models produce quasi-periodic characteristic earthquakes, their behavior is qualitatively similar to lower-dimension models. Certain coseismic characteristics like stress drop and fracture energy are largely controlled by frictional parameters and are thus largely comparable. However, other observations are quantitatively clearly affected by dimension reduction. We find corresponding increases in recurrence interval, coseismic slip, peak slip velocity, and rupture speed. These changes are to a large extend explained by the elimination of velocity-strengthening patches that transmit tectonic loading onto the velocity-weakening fault patch, thereby reducing the interseismic stress rate and enhancing the slip deficit. This explanation is supported by a concise theoretical framework, which explains some of these findings quantitatively and effectively estimates recurrence interval and slip. Through accounting for an equivalent stressing rate at the nucleation size h* into 2/3D models, 0/1D models can also effectively estimate these earthquake cycle parameters. Given the computational efficiency of lower-dimensional models that run more than a million times faster, this paper aims to provide qualitative and quantitative guidance on economical model design and interpretation of modeling studies.

Dynamic modeling of sequences of earthquakes and aseismic slip (SEAS) provides a self-consistent, physics-based framework to connect, interpret, and predict diverse geophysical observations across spatial and temporal scales. Amid growing applications of SEAS models, numerical code verification is essential to ensure reliable simulation results but is often infeasible due to the lack of analytical solutions. Here, we develop two benchmarks for three-dimensional (3D) SEAS problems to compare and verify numerical codes based on boundary-element, finite-element, and finite-difference methods, in a community initiative. Our benchmarks consider a planar vertical strike-slip fault obeying a rate- and state-dependent friction law, in a 3D homogeneous, linear elastic whole-space or half-space, where spontaneous earthquakes and slow slip arise due to tectonic-like loading. We use a suite of quasi-dynamic simulations from 10 modeling groups to assess the agreement during all phases of multiple seismic cycles. We find excellent quantitative agreement among simulated outputs for sufficiently large model domains and small grid spacings. However, discrepancies in rupture fronts of the initial event are influenced by the free surface and various computational factors. The recurrence intervals and nucleation phase of later earthquakes are particularly sensitive to numerical resolution and domain-size-dependent loading. Despite such variability, key properties of individual earthquakes, including rupture style, duration, total slip, peak slip rate, and stress drop, are comparable among even marginally resolved simulations. Our benchmark efforts offer a community-based example to improve numerical simulations and reveal sensitivities of model observables, which are important for advancing SEAS models to better understand earthquake system dynamics.

The theory of rate and state friction unifies field, laboratory, and theoretical analysis of the evolution of slip on natural faults. While the observational study of earthquakes and aseismic fault slip is hampered by its strong multi-scale character in space and time, numerical simulations are well-positioned to link the laboratory study of grain-scale processes to the scale at which rock masses move. However, challenges remain in accurately representing the complex and permanently evolving sub-surface fault networks that exist in nature. Additionally, the common representation of faults as interfaces may miss important physical aspects governing volumetric fault system behavior. In response, we propose a transient viscous rheology that produces shear bands that closely mimic the rate- and state-dependent sliding behavior of equivalent fault interfaces. Critically, we show that the expected tendency of the continuum rheology for runaway localization and mesh dependence can be halted by including an artificial diffusion-type regularization of anelastic strain rate in the softening law. We demonstrate analytically and numerically using a simplified fault transect that important aspects of the frictional behavior are not significantly affected by the introduced regularization. Any discrepancies with respect to the interfacial description of fault behavior are critically evaluated using 1D numerical velocity stepping and spring-slider experiments. ;Since no new physical parameters are introduced, our model may be straightforwardly used to extend the existing modeling techniques. The model predicts the emergence of complex patterns of shear localization and delocalization that may inform the interpretation of complex damage distributions observed around faults in nature.