Natural fault networks are geometrically complex systems that evolve through time. The evolution of faults and their off-fault damage patterns are influenced by both dynamic earthquake ruptures and aseismic deformation in the interseismic period. To better understand each of their contributions to faulting we simulate both earthquake rupture dynamics and long-term deformation in a visco-elasto-plastic crust subjected to rate- and state-dependent friction. The continuum mechanics-based numerical model presented here includes three new features. First, a 2.5-D approximation is created to incorporate the effects of a viscoelastic lower crustal substrate below a finite depth. Second, we introduce a dynamically adaptive (slip-velocity-dependent) measure of fault width to ensure grid size convergence of fault angles for evolving faults. Third, fault localization is facilitated by plastic strain weakening of bulk rate and state friction parameters as inspired by laboratory experiments. This allows us to simulate sequences of episodic fault growth due to earthquakes and aseismic creep for the first time. Localized fault growth is simulated for four bulk rheologies ranging from persistent velocity weakening to velocity strengthening. Interestingly, in each of these bulk rheologies, faults predominantly localize and grow due to aseismic deformation. Yet, cyclic fault growth at more realistic growth rates is obtained for a bulk rheology that transitions from velocity-strengthening friction to velocity-weakening friction. Fault growth occurs under Riedel and conjugate angles and transitions towards wing cracks. Off-fault deformation, both distributed and localized, is typically formed during dynamic earthquake ruptures. Simulated off-fault deformation structures range from fan-shaped distributed deformation to localized splay faults. We observe that the fault-normal width of the outer damage zone saturates with increasing fault length due to the finite depth of the seismogenic zone. We also observe that dynamically and statically evolving stress fields from neighboring fault strands affect primary and secondary fault growth and thus that normal stress variations affect earthquake sequences. Finally, we find that the amount of off-fault deformation distinctly depends on the degree of optimality of a fault with respect to the prevailing but dynamically changing stress field. Typically, we simulate off-fault deformation on faults parallel to the loading direction. This produces a 6.5-fold higher off-fault energy dissipation than on an optimally oriented fault, which in turn has a 1.5-fold larger stress drop. The misalignment of the fault with respect to the static stress field thus facilitates off-fault deformation. These results imply that fault geometries bend, individual fault strands interact, and optimal orientations and off-fault deformation vary through space and time. With our work we establish the basis for simulations and analyses of complex evolving fault networks subject to both long-term and short-term dynamics.
Immature strike-slip faults accumulate displacement over time as they undergo a slip localization process. In the long term, these structures can become deeply penetrating, major faults that represent a highly localized weak zone through the lithosphere
Fault structures developing at the tip of a sliding strike-slip fault. Modified from
The secondary fault structures altogether constitute the wake of a permanent off-fault damage zone
Initial attempts to simulate plastic off-fault deformation in elastodynamic earthquake mechanics models were undertaken by
In this paper we develop a computational model that combines the following features: dynamic off-fault yielding in a visco-elasto-plastic material, long-term evolution of a geometrically complicated fault system, consistent simulation of multiple subsequent earthquakes on the same fault system and the effect of the finite seismogenic–elastic depth.
Our method builds upon and extends the recently developed STM-RSF numerical model for seismo-thermomechanical modeling under rate and state friction
To accurately simulate cyclic fault growth with off-fault plasticity we extend this STM-RSF framework with three new features.
First, we compare four different rate-dependent rheologies in the bulk material. The most realistic one is inspired by laboratory friction experiments and includes a weakening of RSF parameters with plastic strain
We summarize the main ingredients of the STM-RSF modeling approach in Sect.
We expand the 2-D STM-RSF framework to 2.5-D using a generalized Elsasser approach introduced by
In 2.5-D we solve for the conservation of mass,
The local effective friction parameter
Localized bulk deformation and fault slip are related by defining the plastic slip rate
where
To solve the governing equations we use an implicit, conservative, finite-difference scheme on a fully staggered grid combined with the marker-in-cell technique
In contrast to previous studies, in which fault width
We propose a new invariant continuum-based RSF formulation, in which the fault width
The dimensionless scaling parameter
The result of using the new slip-velocity-dependent fault-width formulation is that both the fault angle and the onset times of earthquakes converge with grid size (Appendix
Rate- and state-dependent friction (RSF) and material parameters of the four 2.5-D reference models.
Our 2.5-D model setup represents a generic case to study the evolution of a fault zone within a plane strain crust of 20
2.5-D model setup of the dextral in-plane strike-slip simulation: 2-D box of size
The maximum compressive stress
Considering a typical value of the pore fluid pressure ratio
In this study we present four different reference models with varying RSF behavior in the bulk:
model RS has rate-strengthening behavior,
model RN has rate-neutral behavior,
model RW has rate-weakening behavior,
and model RT has a transition from rate-strengthening to rate-weakening behavior at increasing plastic strain.
We choose to study these different bulk behaviors because in the literature of materials and geology both strain-rate strengthening and strain-rate weakening are reported to be possible and sufficient conditions for localization of deformation
In the first part of this section we present and compare the results of the four models to understand the effects of a rate-sensitive bulk rheology on off-fault deformation and fault growth.
We then focus on two reference models to further investigate factors influencing the off-fault fan width and the role of viscoelastic lower crust relaxation in short-term and long-term off-fault deformation (Sect.
The initial elastic loading phase takes 325.5 years in all four reference models. In this initial stage stresses start to localize on the predefined fault and its slip velocity increases exponentially (Fig.
Besides these general similarities among the four reference models, we note two main differences.
First, the plastic zones off the main fault have distinct characteristics. A fan of diffuse deformation occurs in models RS and RT, while localized deformation on secondary faults occurs in models RN and RW.
Thus, the type of plastic off-fault yielding depends on the properties of the bulk rheology.
Only a rate-neutral or rate-weakening material in which
The second main difference is that only model RT hosts a sequence of several earthquakes. In the other three models the entire new fault geometry forms during a single earthquake, an aftershock, and the subsequent post-seismic and interseismic phases. Before more seismic events can nucleate, the main fault has already extended by aseismic growth toward a model boundary and the simulation is stopped to prevent boundary effects.
The models with constant rate sensitivity (models RS, RN and RW) have fast fault growth rates of up to 77
Summary of four different right-lateral reference models.
In the following analysis we mainly focus on models RW and RT because the off-fault characteristics of models RW and RN are similar, and those of models RT and RS. Furthermore, RW and RT represent the two cases of dominant higher-angle and lower-angle continuing faults, respectively. In that way, RW and RT are the most diverse end-member cases of the four reference models. At the same, time model RT is the most realistic model with successive earthquakes, distributed deformation and localized fault growth. Model RW, on the other hand, allows for strong off-fault localization and tends towards irregular fault patterns with unevenly spaced secondary fault branches.
As the main fault propagates beyond the tip of the predefined fault, all four reference models form two new faults with two orientations.
A “higher-angle fault” forms with a high angle compared to the strike of the predefined fault. It is an antithetic conjugate Riedel shear fault and is termed R*2, wherein “*” stands for the different reference models. A “lower-angle fault” forms with a lower strike angle and is a synthetic Riedel shear fracture termed R*1 hereafter.
Detailed analysis of model RT reveals that these two faults are formed because the main fault rupture induces two stress lobes on the extensional side of the fault. These zones were studied by
Evolution of the local friction coefficient and stress orientation of model RT over time and the prediction of the absolute fault angle. The three different times are (1) 5 s before fault propagation (symbol
The different reference models favor either of the two fault angle types or develop them jointly.
The smaller the initial
In the following we describe the evolution of the R1 fault and the conjugate R2 in detail for the four individual model cases (link to video RT, 116 Mb,
In the rate-neutral model the propagating rupture excites several equidistant secondary splay faults that form under the same Coulomb angle (R1) and saturate at
In the rate-weakening model RW, the secondary splay faults form earlier than in model RN and are more localized and partly non-equidistant. Particularly apparent is a secondary splay at
The off-fault deformation pattern in the rate transition model is composed of similar features as model RS. The two branches RT1 and RT2 form as the main fault rupture penetrates into the undeformed bulk.
The fault evolution in model RT spans several earthquake cycles (Fig.
In summary, the strike angle of the formed faults increases from rate-strengthening to rate-neutral to rate-weakening material. The case of a transition from RS to RW describes a more complex case with intermittent aseismic and seismic growth, fault bends, fault interaction and additional off-fault deformation. In addition, the degree of new fault localization is highest in model RW and follows the order RW, RN, RT and RS.
Temporal and spatial evolution of model RT with indications of aseismic and seismic fault growth stages in color.
In this section we first analyze the properties of the saturating plastic off-fault fan and study the determining parameters for this saturation. Secondly, we study the long-term effects of the relaxed viscoelastic substrate on the growth of the splay fault fan.
In all four reference models the plastic off-fault fan reaches a maximal width
Accumulated plastic slip and slip velocity during the first earthquake on the main fault plotted in 5 s intervals.
We investigate the impact of crustal thickness
We additionally vary the initial background pressure
Impact of various parameters on the maximum plastic fan width
In Sect.
Comparison of plastic off-fault strain and energy between model RT and OOF (optimally oriented fault model).
In Fig.
In order to increase the strong off-fault localization of reference model RW, with its propensity towards irregular fault patterns and unevenly spaced secondary fault branches, we increase the initial background pressure
In the following we analyze several indications of fault and rupture interactions due to stress changes that are typically ignored in seismic cycle models.
They include (1) rupture arrest when two subparallel ruptures get too close to one another. This can be observed for fault HPT001, which stops growing because the stresses on the extensional side of the subsequently forming branch HPT01 increase, become dominant and limit the compressional side stresses of HPT001. As a consequence, only extensional stresses remain at the tip of HPT001 such that the fault becomes thinner on its compressional side (Fig.
Model HPT with higher background pressure (
The main fault rupture forms a Riedel fault HPT1 and a conjugate HPT2 like in model RW.
These two faults and the secondary branch HPT01 grow until the event slip velocity drops to
Our results suggest that all four reference models have predominantly aseismically growing faults.
A bulk rheology with constant rate sensitivity favors faster fault growth. In contrast, the heterogeneities introduced by a weakening of the RSF parameters
The concept of work minimization states that new faulting starts when the active fault has become suboptimal in the Coulomb sense and inefficient, with sufficiently high amounts of strain transferred into the surrounding rock
Interestingly, most of the dynamically generated Riedel faults are abandoned after they form. An exception is the Riedel fault at
We analyzed the angles of newly formed Riedel faults R1 and R2 and showed that they comply with the Mohr–Coulomb faulting theory.
Earthquakes on the main fault induce a dynamic elevation of the local stress and friction coefficient and a lobe-like alteration of the stress orientations.
These dynamic changes determine, via classical failure theory, greater fault angles than the typical 10–
Owing to the differences between quasi-static and dynamic stress and strength conditions, the faults in our models reorient and bend as they alternate between aseismic and seismic growth stages.
The fault angle
To summarize, our findings imply that fault bending is most likely the result of a misalignment of the preexisting fault, which can occur also in a frictionally homogeneous medium. This fault misalignment can be strongly affected by seismically activated dynamic processes. Fault bending must not necessarily be the result of only seismic rupturing, but the magnitude of bending can be strongly increased by it. Additionally, modeling shows that bending related to seismic rupture smears out over time, but an overall increase in the angle of the entire fault trace can be recorded in the long term.
All four reference models agree to first order with the finding that the maximum amount a fault can grow in a single earthquake that ruptures the entire fault is of the order of 1 % of its previous length
Our study shows that the amount of off-fault deformation is crucially dependent on the misalignment of the fault or, in other words, on the optimality of the angle of a predefined fault (Sect.
Our findings regarding the time-dependent optimality of the fault angle have implications for nature and for future dynamic rupture modeling studies.
Active fault strands in nature that are surrounded by severe localized or diffuse damage zones, possibly extending far into the host rock, are strongly misaligned with the interseismic far-field stress field. This misalignment may be increased dynamically during seismic rupturing.
This means that individual fault traces may reflect the local geology, structure or stress state rather than the prevailing far-field long-term stress field, and this effect would vary from segment to segment, randomizing the fault pattern
These statements are supported by model HPT, wherein strong local alterations of stresses lead to marked secondary rupturing and a main fault replacement (Fig.
It is noted that the typical dynamic rupture modeling setup of a
We found evidence for connecting fault segments, which highlights the fact that different fault strands interact both during and between earthquakes. Dynamic interaction is versatile and pronounced in model HPT (Fig.
An interesting feature in model HPT is the main fault replacement (or jump).
This is reflected in the singular growth and slip activity of the outermost fault branch HPT0001 at the end of the simulation. In the dynamically altered stress field this outermost fault branch is most favorably oriented.
A main fault jump was reported in southern California where the San Gabriel Fault was originally the main strand of the San Andreas Fault but was replaced at about 4 Ma
Fault branch interaction occurs also in the long term when the stress fields of approaching fault strands start to interfere (manifest in a seismically initiated incipient connection between RT1 and RT2 at
The decreasing strike angle of fault RT2 brings another aspect with it. The fault RT2 is initially formed at the typical angle of a conjugate Riedel fault (R
Such classical wing cracks are typically found in laboratory experiments and as subsidiary cracks in nature (e.g.,
In Sect.
The natural faulting process is a three-dimensional process. Compared to previous studies that applied the STM code, we approach three dimensionality here with a 2.5-D approximation. We thus obtain a finiteness of the seismogenic depth that limits the stress concentration at the fault tip, which in turn limits the spatial extent of plasticity outside the main fault
In this study we simulated the spatiotemporal evolution of a complex strike-slip fault system subjected to repeated earthquake ruptures.
We applied an invariant rate- and state-dependent friction formulation framework that allows for the spontaneous growth and evolution of a fault.
This STM-RSF framework was extended with a 2.5-D approximation, a new dynamically adapting slip-velocity-dependent fault-width formulation and a plastic-strain-weakening mechanism of bulk parameters inspired by laboratory experiments.
With this advanced model we present different possibilities of how a strike-slip fault grows due to (a)seismic processes in different host rock rheologies of which the end-member cases are bulk velocity weakening and bulk velocity strengthening.
This work focuses on three main aspects. It discriminates between the conditions leading to distributed or localized dynamic off-fault deformation and the saturation of the plastic zone width. Our models distinguish between off-fault deformation geometries observed in nature (Fig. This study analyzes distinct propagation styles of the main fault leading to a complex interactive fault network with bends caused by differences in angle between seismic and aseismic segments. The different fault branches are successfully linked to the Mohr–Coulomb faulting criterion. The development of Riedel shear faults and their conjugates is caused by dynamic stress field effects and also explained by the theoretical faulting criterion. Ultimately, our study demonstrates that the amount of plastic off-fault deformation crucially depends on both the initial fault orientation with respect to the far-field stresses and also on the dynamic optimality of the fault angle in relation to local stresses. The optimality of fault alignment in a stress field is time-dependent and depends on local variations of rotating stress orientations.
Additionally, we found that under the wide range of conditions explored the contribution of seismic fault growth is limited compared to the aseismic contribution. Earthquakes rather lead to greater localization in areas of distributed deformation close to the fault tip. Nevertheless, the overall fault angle of a fault that extends by combined aseismic and seismic growth is 14.2
With respect to fault branch and rupture interactions we reported rupture arrest, fault bending, fault convergence and intersection, arrest of fault growth, and fault strand abandoning. Fault interaction was observed in the long term and during coseismic events. In an extensional fault setting the extensional side fault of two subparallel faults is the favored one and likely to continue. The dynamic rotation of stresses can lead to a reorientation of stresses, which might result in the severe misalignment of the former main fault. This will lead to a replacement of the main fault trace and a jump of fault activity. Thus, fault systems tend to optimize their efficiency by adapting to changing conditions. We additionally found that fault systems optimize their growth efficiency by progressively favoring similar growth directions for seismic and aseismic growth.
With our work we provide the basis for simulations and analyses of complex evolving fault networks subject to long-term and short-term dynamics. The approach we presented has potential to be applied to a more realistic fault map in a future study.
We tested the new slip-velocity-dependent fault-width formulation (Eq.
Convergence analysis of fault angle and earthquake time as a relation between grid resolution and fault angles and between grid resolution and earthquake time start, respectively. The analysis shows that all seismic and aseismic fault angles as well as earthquake timing converge with grid size. Color-filled symbols indicate different faulting stages. Blue error bars correspond to errors in measuring the absolute angle
The code is available upon request to Taras Gerya (taras.gerya@erdw.ethz.ch). With this code the four reference models can be rerun. Figures
The repository cited in the references (
SP designed the study, implemented and tested the 2.5-D approximation and the dynamically adaptive measure of fault width, designed the model setup, decided the parameter space, ran the simulations, gathered and interpreted the results, wrote the paper, and organized the submission process. JPA provided the initial idea to use the 2.5-D generalized Elsasser approach, helped clarify the results and improved the paper. TG provided the initial idea and tests of the dynamically adaptive measure of fault width, assisted in formulating the 2.5-D approximation for STM-RSF, and helped to design the initial model setup. YvD helped clarify the results and improved the paper.
The authors declare that they have no conflict of interest.
We thank Robert Herrendörfer for providing the STM-RSF code.
We additionally thank André Niemeijer for a discussion on rate and state friction parameters for the bulk and Jianye Chen for giving very useful comments on the slip-velocity-dependent fault-width formulation.
For constructive comments and discussions we thank the STM group at ETH Zurich.
Numerical simulations were performed on the ETH clusters Leonhard and Euler.
Perceptually uniform color maps are used in this study to prevent visual distortion of the data
This research has been supported by the Swiss National Science Foundation SNF (grant no. 200021_182069). Jean Paul Ampuero was supported by the French National Research Agency (ANR) through project FAULTS_R_GEMS (grant ANR-17-CE31-0008) and the Investments-in-the-Future project UCAJEDI (grant ANR-15-IDEX-01).
This paper was edited by David Healy and reviewed by Michele Cooke and Boris Kaus.