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# Cascade of weakening at the tip of a dynamic shear rupture. a, Schematic of a slip-pulse mode of rupture: re-locking follows fault rupture during rupture propagation. b, Evolution of the fault slip rate and strength at the rupture tip (for example, shaded region in a), with the parameter set in Supplementary Table 2. Initial weakening occurs through reduction of the friction coefficient by flash weakening. Subsequently, thermal pressurization of fault gouge pore fluid reduces the effective normal stress. Two endmembers are considered: negligible (cyan lines) and non-negligible (red lines) flux of heat and pore fluid. Broken lines follow the evolution under flash weakening of friction without thermal pressurization (dashed line) or thermal pressurization under a flash-weakened coefficient of friction (dotted and dash-dotted lines).

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Laboratory simulations of earthquakes show that at high slip rates, faults can weaken significantly, aiding rupture [1–3] . Various mechanisms, such as thermal pressurization and flash heating, have been proposed to cause this weakening during laboratory experiments [1,4–6], yet the processes that aid fault slip in nature remain unknown. Measuremen...

## Contexts in source publication

**Context 1**

... source parameters (Methods and Supplementary Tables 3-5 and Supplementary Figs 1-3), namely the total moment and seismic energy released and a characteristic source radius. On the basis of seismic source spectra, for each event, we estimate an intermediate energetic quantity G = ((τ/2 − τ a )δ (where τ is the static stress drop and τ a is the apparent stress). ...

**Context 2**

... the off-fault contribution to the fracture energy G off ≈ (0.1-10) MJ m −2 (δ/1 m), where the lower limit roughly corresponds to the case of low fault shear stress inferred on major plate-bounding faults 95 . This estimate is approximately an order of magnitude below the fracture energy estimates from seismological observations for larger events (Fig. 3, δ 0.1 m). Although we caution that this is a preliminary estimate, this suggests that on-fault contributions to fracture energy dominate when fault strength weakens slowly with slip, as for weakening by thermal ...

**Context 3**

... observation pertains to the question how well do the 'local' G versus slip data inferred from a kinematic inversion agree with the fault-averaged data. We provide an illustration of such an agreement in Supplementary Fig. 1, where we plot the local fracture energy versus slip data (cyan stars) from selected locations in the finite-fault model of the Imperial Valley event 96 together with the fault-average values for other large crustal earthquakes (as in Fig. 3c). Evidently, the local fault data scales in approximately the same way as the averaged data from different events (including the fault-average value for the Imperial Valley event itself). ...

**Context 4**

... availability. The approach used to produce the theoretical curves in Figs 1-3 is detailed in the Methods. The resulting algebraic equations are easily solved using a Newton-Raphson procedure, making inclusion of the underlying code unnecessary. ...

**Context 5**

... range of the relative value of the slip-weakening distances, L ⇤ / c , is determined by the "intact-thick" and "damaged-thin" configurations, which respectively give the smallest and largest values. Table S.2: Model parameters used to determine solutions appearing in Figures 1-3. Figure 1b cyan 10 3 0 red 10 3 10 dashed 10 3 a n/a dotted n/a 0 dash-dotted - 10 b Table S.3: Megathrust subduction earthquakes source parameters from compilations of events from years '01-'11 ( Lay et al. 35 ), years '94-'96 (Venkataraman & Kanamori 36 ), and year '85 (Choy & Dewey 37 ). ...

**Context 6**

... S.2: Model parameters used to determine solutions appearing in Figures 1-3. Figure 1b cyan 10 3 0 red 10 3 10 dashed 10 3 a n/a dotted n/a 0 dash-dotted - 10 b Table S.3: Megathrust subduction earthquakes source parameters from compilations of events from years '01-'11 ( Lay et al. 35 ), years '94-'96 (Venkataraman & Kanamori 36 ), and year '85 (Choy & Dewey 37 ). seismic moment; r, e↵ective source radius evaluated from the fault area with significant slip in the corresponding kinematic slip inversion for all main shocks, and from signal durations for the Chile '85 aftershocks 37 ; ⌧ i , prestress estimates of a Lamb 73 , b Seno 74 , and the geometric mean of the two (where applicable); ⌧ = ⌧ i ⌧ f = 7M 0 /16r 3 , stress drop; = M 0 /µ⇡r 2 , average slip calculated using µ = 40 GPa (µ = 65 GPa for deeper Pagai Is., Benkulu, Miyagi, and Chile'85 mainshocks, and Chile'85 aftershocks 2 and 8); E s , radiated energy estimates for events in years '01-'11 35 , for '94-'96 36 , and for Chile '85 Main Shock and aftershock sequence 37 ; G 0 = (⌧ /2 ⌧ a ) with ⌧ a = E s /⇡r 2 and G max = G 0 + (⌧ i + ⌧ ), fracture energy end-members 4 Mo, seismic moment; r, e↵ective source radius calculated for the fault area with significant slip (> 33% of the mean subfault slip) in the corresponding kinematic slip inversion; ⌧ = ⌧ i ⌧ f = 7M 0 /16r 3 , stress drop; = M 0 /µ⇡r 2 , average slip calculated using µ = 30 GPa; Es, radiated energy from a regional, b teleseismic, and otherwise the geometric-mean of the regional and teleseismic estimates (when available). ...

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We present a model for dynamic weakening of faults based on local flash heating at microscopic asperity contacts coupled to bulk heating at macroscopic scale. We estimate the fracture energy G associated with that rheology, and find that for constant slip rate histories G scales with slip δ as G ~ δ^2 at small slip, while G ~ δ^1/2 at large slip. T...

Theoretical studies predict that during earthquake rupture faults slide at non-constant slip velocity, however it is not clear which source time functions are compatible with the high velocity rheology of earthquake faults. Here we present results from high velocity friction experiments with non-constant velocity history, employing a well-known sei...

## Citations

... To compare our modeling results with existing measurements of fracture energy from earthquakes globally, we compile fracture energy estimates made previously with events spanning five orders of magnitude in size, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005) (Fig. 10). Previous inferences have shown a nonlinear scaling of fracture energy with slip (Abercrombie and Rice, 2005), from G ∝ δ 2 for small earthquakes to G ∝ δ 2/3 for large earthquakes (Fig. 10), which has been attributed to thermal pressurization of pore-fluid by the rapid shear heating of fault gouge (Viesca and Garagash, 2015). ...

... To compare our modeling results with existing measurements of fracture energy from earthquakes globally, we compile fracture energy estimates made previously with events spanning five orders of magnitude in size, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005) (Fig. 10). Previous inferences have shown a nonlinear scaling of fracture energy with slip (Abercrombie and Rice, 2005), from G ∝ δ 2 for small earthquakes to G ∝ δ 2/3 for large earthquakes (Fig. 10), which has been attributed to thermal pressurization of pore-fluid by the rapid shear heating of fault gouge (Viesca and Garagash, 2015). For the set of parameters considered in this study, our models produce both relatively large stress drop (τ s − τ d ) and large characteristic slip weakening distance (d c ), which results in high values of fracture energy (Fig. 9c). ...

... However, ample observations suggest Fig. 10. Compilation of fracture energy (G ′ ) and slip (δ) from different earthquakes worldwide, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005). Black dashed lines indicate the different scaling for small (G ∝ δ 2 ) and large (G ∝ δ 2/3 ) earthquakes. ...

A major goal in earthquake physics is to derive a constitutive framework for fault slip that captures the dependence of shear strength on fault rheology, sliding velocity, and pore-fluid pressure. In this study, we present H-MEC (Hydro-Mechanical Earthquake Cycles), a newly-developed two-phase flow numerical framework — which couples solid rock deformation and pervasive fluid flow — to simulate how crustal stress and fluid pressure evolve during the earthquake cycle. This unified, continuum-based model, incorporates a staggered finite difference–marker-in-cell method and accounts for inertial wave-mediated dynamics and fluid flow in poro-visco-elasto-plastic compressible medium. Global Picard-iterations and an adaptive time stepping allow the correct resolution of both long- and short-time scales, ranging from years to milliseconds. We present a comprehensive in-plane strike-slip setup in which we test analytical poroelastic benchmarks of pore-fluid pressure diffusion from an injection point along a finite fault width. We then investigate how pore-fluid pressure evolution and solid–fluid compressibility control sequences of seismic and aseismic fault slip. While the weakening phase is controlled by localized compaction of pores and dynamic self-pressurization of fluids inside the undrained fault zone, the subsequent propagation of dynamic ruptures is driven by pore-pressure waves. Furthermore, pore-fluid pressure conditions on the fault and shear strength weakening associated with rapid self-pressurization of fluids control the characteristic slip-weakening distance, the final size of seismic events, and the scaling between slip and fracture energy observed for large earthquakes. Our modeling results demonstrate that fault failure can occur due to poroelastic coupling on a finite-width shear zone, thus highlighting the importance of considering the realistic hydro-mechanical structure of faults to investigate fluid-driven seismic and aseismic slip, either as a natural process or induced by human activities.

... Our results demonstrate that detailed analysis of gouge fabric orientation is an effective method for inferring the post-earthquake stress state on a slip plane that experiences rapid slip. Because thermal pressurization is recognized to be a potentially widespread phenomenon that takes place along seismic faults during earthquakes (Viesca and Garagash, 2015), we expect that similar observations could be made elsewhere to infer post-earthquake stress orientations on faults. A key advantage of using gouge fabric over existing borehole methods is that the stress orientation can be inferred directly on the principal slip plane, which is usually no thicker than a few centimeters and is unresolvable using techniques like hydraulic fracturing, observation of wellbore failures, ASR, and overcoring. ...

Stress on seismogenic faults provides critical information about how much elastic energy is stored in the crust and released by earthquakes, which is crucial in understanding earthquake energetics and recurrence. However, determining post-earthquake stress states on faults remains challenging because current borehole methods are rarely applicable to damaged fault zone rocks. We applied neutron texture analysis to gouge samples of the 1999 Chi-Chi earthquake in Taiwan to infer the stress state after the earthquake. Results indicate that the clay fabric within the principal slip zone is orthogonal to the fault plane, whereas outside the principal slip zone the fabric is predominantly parallel to the bedding-parallel fault plane. We suggest that the clay fabric in the slip zone was first neutralized by the coseismic fluidization caused by thermal pressurization and later re-oriented to the new direction of post-earthquake principal stress. Such stress orientation is consistent with the orientations inferred from core-scale fault slip data and dislocation models constrained from global navigation satellite system displacements. If thermal pressurization is a ubiquitous process during earthquakes, gouge fabrics can be used to help probe the post-earthquake stress state of faults.

... For this reason, breakdown work is often estimated from kinematic models with limited frequency bands, or constraining a priori a given weakening law, with the possibility of influencing the final values. With this in mind, such estimates indicate that 'seismological' fracture energy scales with earthquake slip, as a power law with an exponent ranging from 0.5 to 2 (e.g., Abercrombie and Rice, 2005;Viesca and Garagash, 2015). ...

... By contrast with laboratory rupture experiments, friction experiments at high slip velocity, aimed at characterizing the evolution of frictional strength that would be observed at a single point along the fault during seismic slip and have reproduced the slipdependence of breakdown work, with values ranging between 1 kJ/m 2 to 10 MJ/m 2 Cornelio et al., 2020;Seyler et al., 2020;Passelègue et al., 2016). Similarly, fault models based on weakening mechanisms such as thermal pressurization (Viesca and Garagash, 2015;Lambert and Lapusta, 2020) or flash heating (Brantut and Viesca, 2017) have also been shown to exhibit such scaling between slip and breakdown work. In both friction experiments and models, most of the total dissipated energy is converted into frictional dissipation, further enhancing the weakening of the fault during coseismic slip due to the occurrence of thermally activated weakening processes. ...

... In consequence, the amount of breakdown work generated during rupture propagation would be the result of the final slip and of the initial shear stress acting along the fault (i.e. of the stress drop), rather than the final rupture length (as for a circular crack model). Of course, in nature, the evolution of stress with slip is expected to deviate from the simple linear slip-weakening behavior assumed for simplicity in this work, which is expected to modify the slip dependence of the breakdown work, as observed in recent studies (Viesca and Garagash, 2015;Lambert and Lapusta, 2020). However, the activation of different weakening mechanisms with increasing slip suggests that while natural earthquakes might be expected to initiate like classical shear cracks, subsequent frictional weakening at the scale of the entire rupture can help the earthquake to grow further into lowly-stressed regions of the fault and across barriers. ...

Potential energy stored during the inter-seismic period by tectonic loading around faults is released during earthquakes as radiated energy, frictional dissipation and fracture energy. The latter is of first importance since it is expected to control the nucleation, the propagation and the arrest of the seismic rupture. On one side, the seismological fracture energy estimated for natural earthquakes (commonly called breakdown work) ranges between 1 J/m² and tens of MJ/m² for the largest events, and shows a clear slip dependence. On the other side, recent experimental studies highlighted that, concerning rupture experiments, fracture energy is a material property (energy required to break the fault interface) independently of the size of the event, i.e. of the seismic slip. To reconcile these contradictory observations and definitions, we performed stick-slip experiments, as analog for earthquakes, in a bi-axial shear configuration. We estimated fracture energy through both Linear Elastic Fracture Mechanics (LEFM) and a Cohesive Zone Model (CZM) and through the integration of the near-fault stress-slip evolution. We show that, at the scale of our experiments, fault weakening is divided into a near-tip weakening, corresponding to an energy of few J/m², consistent with the one estimated through LEFM and CZM, and a long-tailed weakening corresponding to a larger energy not localized at the rupture tip, increasing with slip. Through numerical simulations, we demonstrate that only near-tip weakening controls the rupture initiation and that long-tailed weakening can enhance slip during rupture propagation and allow the rupture to overcome stress heterogeneity along the fault. We conclude that the origin of the seismological estimates of breakdown work could be related to the energy dissipated in the long-tailed weakening rather than to the one dissipated near the tip.

... However, these properties are difficult to measure in the Earth. A number of studies tried to constrain fault constitutive behavior using seismological observations of earthquakes, and, in particular, the way that earthquake parameters scale from small to large [1][2][3][4] . One significant seismological observation, known as the breakdown energy, is thought to be related to the slip weakening process, and is often assumed to be a proxy for fracture energy. ...

... One significant seismological observation, known as the breakdown energy, is thought to be related to the slip weakening process, and is often assumed to be a proxy for fracture energy. However, the breakdown energy is frequently observed to be negative-valued 1,4,5 and, if positivevalued, to scale with slip. Hence, larger earthquakes with more total slip appear to dissipate more energy (per unit rupture area) than small earthquakes. ...

... For instance, it was suggested that frictional weakening distance could increase with increasing earthquake size 1 . Other studies suggested additional mechanisms that activate as the earthquake rupture grows larger and slip distances increase such as thermal pressurization (e.g., ref. 3,4,9,10 ) or off-fault energy sinks (e.g., ref. [11][12][13][14]. High-velocity friction experiments that show continued weakening with cumulative slip were also offered as support. ...

In the quest to determine fault weakening processes that govern earthquake mechanics, it is common to infer the earthquake breakdown energy from seismological measurements. Breakdown energy is observed to scale with slip, which is often attributed to enhanced fault weakening with continued slip or at high slip rates, possibly caused by flash heating and thermal pressurization. However, seismologically inferred breakdown energy varies by more than six orders of magnitude and is frequently found to be negative-valued. This casts doubts about the common interpretation that breakdown energy is a proxy for the fracture energy, a material property which must be positive-valued and is generally observed to be relatively scale independent. Here, we present a dynamic model that demonstrates that breakdown energy scaling can occur despite constant fracture energy and does not require thermal pressurization or other enhanced weakening. Instead, earthquake breakdown energy scaling occurs simply due to scale-invariant stress drop overshoot, which may be affected more directly by the overall rupture mode – crack-like or pulse-like – rather than from a specific slip-weakening relationship. Earthquake breakdown energy is commonly interpreted as a proxy for fracture energy but is observed to scale with magnitude. Here the authors show that a scale-independent stress overshoot, as seen in the 3D dynamic earthquake rupture simulations, leads to comparable scaling despite constant fault fracture energy.

... Favourably oriented frictional weakening faults with respect to the in-situ stress field, typically characterised by a large stress criticality ( s o s p .1), are very susceptible to host run-away seismic ruptures. Indeed, a little stress perturbation is sufficient to re-activate slip and its velocity propagation tends to diverge rapidly due to friction weakening and possibly other weakening mechanisms, such us flash-heating and thermal pressurization (Viesca and Garagash 2015). Garagash and Germanovich (2012) have shown via a stability analysis that critically stressed pressurized faults, for which the relation s o [ s r ¼ f r r 0 o is strictly satisfied, host always the nucleation of an unabated dynamic event. ...

... Thermal pressurization of pore-fluid by rapid shear heating of fault gouge unit has been showed to be a prominent process of fault weakening (Viesca and Garagash 2015). This mechanism depends on fluid thermodynamics properties and drives the shear strength loss at low fluid pressure conditions (Acosta et al. 2018). ...

Fluid injection into underground formations reactivates preexisting geological discontinuities such as faults or fractures. In this work, we investigate the impact of injection rate ramp-up present in many standard injection protocols on the nucleation and potential arrest of dynamic slip along a planar pressurized fault. We assume a linear increasing function of injection rate with time, up to a given time $$t_c$$ t c after which a maximum value $$Q_m$$ Q m is achieved. Under the assumption of negligible shear-induced dilatancy and impermeable host medium, we solve numerically the coupled hydro-mechanical model and explore the different slip regimes identified via scaling analysis. We show that in the limit when fluid diffusion time scale $$t_w$$ t w is much larger than the ramp-up time scale $$t_c$$ t c , slip on an ultimately stable fault is essentially driven by pressurization at constant rate. Vice versa, in the limit when $$t_c/t_w \gg 1$$ t c / t w ≫ 1 , the pressurization rate, quantified by the dimensionless ratio $$\dfrac{Q_m t_w}{t_c Q^*}$$ Q m t w t c Q ∗ with $$Q^*$$ Q ∗ being a characteristic injection rate scale, does impact both nucleation time and arrest distance of dynamic slip. Indeed, for a given initial fault loading condition and frictional weakening property, lower pressurization rates delay the nucleation of a finite-sized dynamic event and increase the corresponding run-out distance approximately proportional to $$\propto \left( \dfrac{Q_m t_w}{t_c Q^*}\right) ^{-0.472}$$ ∝ Q m t w t c Q ∗ - 0.472 . On critically stressed faults, instead, the ramp-up of injection rate activates quasi-static slip which quickly turn into a run-away dynamic rupture. Its nucleation time decreases non-linearly with increasing value of $$\dfrac{Q_m t_w}{t_c Q^*}$$ Q m t w t c Q ∗ and it may precede (or not) the one associated with fault pressurization at constant rate only.

... To compare our modeling results with existing measurements of fracture energy from earthquakes globally, we compile fracture energy estimates made previously with events spanning five orders of magnitude in size, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005) (Fig. 10). Previous inferences have shown a nonlinear scaling of fracture energy with slip (Abercrombie and Rice, 2005), from G ∝ δ 2 for small earthquakes to G ∝ δ 2/3 for large earthquakes (Fig. 10), which has been attributed to thermal pressurization of pore-fluid by the rapid shear heating of fault gouge (Viesca and Garagash, 2015). ...

... To compare our modeling results with existing measurements of fracture energy from earthquakes globally, we compile fracture energy estimates made previously with events spanning five orders of magnitude in size, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005) (Fig. 10). Previous inferences have shown a nonlinear scaling of fracture energy with slip (Abercrombie and Rice, 2005), from G ∝ δ 2 for small earthquakes to G ∝ δ 2/3 for large earthquakes (Fig. 10), which has been attributed to thermal pressurization of pore-fluid by the rapid shear heating of fault gouge (Viesca and Garagash, 2015). For the set of parameters considered in this study, our models produce both relatively large stress drop (τ s − τ d ) and large characteristic slip weakening distance (d c ), which results in high values of fracture energy (Fig. 9c). ...

... The weakening distance d c has been widely used in fault studies and often imposed as a constant (e.g., Tse and Rice, 1986;Ben-Zion and Rice, 1995;Lapusta et al., 2000;Rubin and Ampuero, 2005;Lapusta and Liu, 2009). However, ample observations suggest that d c should be treated as a variable (Cocco and Bizzarri, 2002;Nielsen et al., 2010), since the effective weakening distance depends on slip history and loading conditions (e.g., Guatteri et al., 2001; Figure 10: Compilation of fracture energy (G ) and slip (δ) from different earthquakes worldwide, from borehole microseismicity to great earthquakes (Abercrombie and Rice, 2005;Rice, 2006;Malagnini et al., 2014;Viesca and Garagash, 2015;Nielsen et al., 2016a;Tinti et al., 2005). Black dashed lines indicate the different scaling for small (G ∝ δ 2 ) and large (G ∝ δ 2/3 ) earthquakes. ...

A major goal in earthquake physics is to derive a constitutive framework for fault slip that captures the dependence of shear strength on fault rheology, sliding velocity, and pore-fluid pressure. In this study, we present H-MEC (Hydro-Mechanical Earthquake Cycles), a newly-developed two-phase flow numerical code - which couples solid rock deformation and pervasive fluid flow - to simulate how crustal stress and fluid pressure evolve during the earthquake cycle on a fluid-bearing fault structure. This unified, continuum-based model, incorporates a staggered finite difference-marker-in-cell (SFD-MIC) method and accounts for full inertial (wave mediated) effects and fluid flow in poro-visco-elasto-plastic compressible medium. Global Picard-iterations and an adaptive time stepping allows the correct resolution of both long- and short-time scales, ranging from years during slow tectonic loading to milliseconds during the propagation of dynamic ruptures. We present a comprehensive in-plane strike-slip setup in which we test analytical poroelastic benchmarks of pore-fluid pressure diffusion from an injection point along a finite fault width. We then investigate how pore-fluid pressure evolution and solid-fluid compressibility control sequences of seismic and aseismic slip on geologic faults. While the onset of fluid-driven shear cracks is controlled by localized collapse of pores and dynamic self-pressurization of fluids inside the undrained fault zone, subsequent dynamic ruptures are driven by solitary pulse-like fluid pressure waves propagating at seismic speed. Furthermore, shear strength weakening associated with rapid self-pressurization of pore-fluid can account for the slip-fracture energy scaling observed for large earthquakes.

... In other words we investigate the conditions inside the fault, when an earthquake happens and a seismic slip takes place. Such an analysis can be used for the calculation of the energy balance, stress drop in the material surrounding the fault as well as defining the conditions under which an earthquake nucleates (see Kanamori & Brodsky, 2004b;Rice, 2006a;Viesca & Garagash, 2015). ...

... In other words, the influence of the boundary conditions during shearing of the fault gouge cannot be ignored. This can be important for experiments in thermal pressurization, calculation of the fracture energy during earthquake nucleation, evaluation of the transition limit from stable to unstable slip, and also for inference of the earthquake spectral characteristics detected in the surface from the fault gouge properties observed in the field (see Aki, 1967;Brune, 1970;Rice, 1973aRice, , 2006aTsai & Hirth, 2020;Viesca & Garagash, 2015). ...

... This implies a partial recovery of the fault strength without it being explicitly implied by the mechanical behavior of the model as in the case of rate and state friction laws (see Dieterich, 1992;Rice et al., 2001;Ruina, 1983a). Therefore, the traveling instability mechanism might work also in the progressive healing of the fault during coseismic slip (see Platt et al., 2014a;Rice, 1973a;Rice et al., 2014a;Viesca & Garagash, 2015). ...

During coseismic slip, the energy released by the elastic unloading of the adjacent earth blocks can be separated in three main parts: The energy that is radiated to the earth’s surface (_ 5% of the whole energy budget), the fracture energy for the creation of new fault surfaces and finally, the energy dissipated inside a region of the fault, with finite thickness, which is called the fault gauge. This region accumulates the majority of the seismic slip. Estimating correctly the width of the fault gauge is of paramount importance in calculating the energy dissipated during the earthquake, the fault’s frictional response, and the conditions for nucleation of the fault in the form of seismic or aseismic slip.In this thesis different regularization approaches were explored for the estimation of the localization width of the fault’s principal slip zone during coseismic slip. These include the application of viscosity and multiphysical couplings in the classical Cauchy continuum, and the introduction of a first order micromorphic Cosserat continuum. First, we focus on the role of viscous regularization in the context of dynamical analyses, as a method for regularizing strain localization. We study the dynamic case for a strain softening strain-rate hardening classical Cauchy continuum, and by applying the Lyapunov stability analysis we show that introduction of viscosity is unable to prevent strain localization on a mathematical plane and mesh dependence.We perform fully non linear analyses using the Cosserat continuum under large seismic slip displacements of the fault gouge in comparison to its width. Cosserat continuum provides us with a proper account of the energy dissipated during an earthquake and the role of the microstructure in the evolution of the fault’s friction. We focus on the influence of the seismic slip velocity to the weakening mechanism of thermal pressurization. We notice that the influence of the boundary conditions in the diffusion of the pore fluid inside the fault gouge, leads to frictional strength regain after initial weakening. Furthermore, a traveling strain localization mode is present during shearing of the layer introducing oscillations in the frictional response. Such oscillations increase the spectral content of the earthquake. Introduction of viscosity in the above mode, leads to a rate and state behavior without the introduction of a specific internal state variable. Our conclusions about the role of thermal pressurization during shearing of the fault gouge, agree qualitatively with newly available experimental results.Finally, based on the numerical findings we investigate the assumptions of the current model of a slip on a mathematical plane, in particular the role of the boundary conditions and strain localization mode in the evolution of the fault’s friction during coseismic slip. The case of a bounded domain and a traveling strain localization mode are examined in the context of slip on a mathematical plane under thermal pressurization. Our results expand the original model in a more general context.

... Importantly, this presence of fluids and thermal variation may give rise to a number of critical phenomena, e.g. thermal pressurization (Rempel & Rice, 2006;Viesca & Garagash, 2015). So the development of a thermo-hydro-mechanical model coupled with the phase-field approaches presented here can be an interesting and meaningful topic. ...

... This energy dissipation can undoubtedly vary the local temperature field, and contribute to some notable phenomena, e.g. flash heating (Viesca & Garagash, 2015). The most critical challenge comes from the complex thermal effects on the constitutive behavior of bulk matrix materials, fracture, and fluid. ...

Geologic materials contain a wide range of discontinuities and fractures, which are central to many engineering applications and geologic hazards. The fracturing process of geologic materials is characterized by its progressive softening, termed quasi-brittleness. Also, the fractured surfaces exhibit a number of features including frictional contact and roughness effects. To model the discontinuities and fracturing processes in geologic materials, the phase-field method has been increasingly applied, as it has an outstanding ability to handle complex crack geometries without using tracking algorithms. However, few phase-field studies considered the quasi-brittleness of geologic materials. More importantly, all existing phase-field approaches dismissed the frictional contact, let alone the roughness effects.
To fill the above research gaps, this thesis develops a suite of phase-field approaches to enable more reliable and systematic modeling of discontinuities and fractures in geologic materials. These approaches focus on four different but interconnected aspects of geologic discontinuities and fractures, namely frictional contact, shear fracture incorporating friction dissipation, mixed-mode rock fracture, and roughness effects of rock fractures. In the first approach, we incorporate the pressure-dependent friction into the phase-field formulation by employing a crack-oriented decomposition of the stress tensor. Each stress component is calculated by identifying the contact condition at the material point of interest. We show that the proposed method can well reproduce the results from the standard and extended finite element method without applying any algorithms to impose contact constraints.
Building on the formulation in the first approach, we develop a phase-field approach to model the shear fracture propagation that involves friction dissipation during the fracturing process. The proposed formulation is demonstrably consistent with a fracture-mechanics-based theory. We also devise a new degradation function to avoid the sensitivity of material strengths to the phase-field length parameters, allowing the proposed method to model quasi-brittle materials with prescribed strengths.
Next, we introduce a double-phase-field approach to the mixed-mode rock fracture by combining the formulations of cohesive tensile cracks and frictional shear cracks. The proposed formulation is essentially based on three steps: (i) stress decomposition in a crack-oriented coordinate system; (ii) calculation of the total potential energy according to the contact condition; (iii) determination of the dominant fracture mode following an energy-based criterion. We validate the double-phase-field approach through qualitative and quantitative comparisons between the modeling results and the experimental results.
Lastly, we introduce a phase-field modeling framework for rock fractures by incorporating roughness effects. The proposed framework aims at transforming a displacement-based constitutive law of rock fractures into a strain-based version without introducing new parameters. In doing so, the continuous phase-field method can accommodate the rough fracture models originally designed for discrete discontinuities. Numerical examples show that the phase-field results have an excellent agreement with the results obtained from the extended finite element method.

... To combine this source mechanism with poroelastic materials, the coupling between poroelastic parameters and parameters of friction laws have to be investigated. To fully capture the interaction of fluids, fault slip and seismic waves, additional multi-physics interactions can be accounted for that describe the thermal pressurisation of pore fluids [79][80][81] during earthquake rupture. The thermal pressurisation model was recently implemented in SeisSol [82]. ...

Many applications from the fields of seismology and geoengineering require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff reactive source term (Darcy's Law) into the elastodynamic wave equations, thereby increasing computational cost of respective numerical solvers and motivating efficient methods utilising High-Performance Computing.
We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary High-Order DERivative time stepping (ADER-DG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we utilise an element-local space-time predictor, which needs to solve medium-sized linear systems – each with 1,000 to 10,000 unknowns – in each element update (i.e., billions of times). We present a novel block-wise back-substitution algorithm for solving these systems efficiently, thus enabling large-scale 3D simulations. In comparison to LU decomposition, we reduce the number of floating-point operations by a factor of up to 25, when using polynomials of degree 6. The block-wise back-substitution is mapped to a sequence of small matrix-matrix multiplications, for which code generators are available to generate highly optimised code.
We verify the new solver thoroughly against analytical and semi-analytical reference solutions in problems of increasing complexity. We demonstrate high-order convergence of the scheme for 3D problems. We verify the correct treatment of point sources and boundary conditions, including homogeneous and heterogeneous full space problems as well as problems with traction-free boundary conditions. In addition, we compare against a finite difference solution for a newly defined 3D layer over half-space problem containing an internal material interface and free surface. We find that extremely high accuracy is required to accurately resolve the slow, diffusive P-wave at a or near a free surface, while we also demonstrate that solid particle velocities are not affected by coarser resolutions. We demonstrate that by using a clustered local time stepping scheme, time to solution is reduced by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis on the SuperMUC-NG supercomputer, demonstrating our implementation's high computational efficiency and its potential for extreme-scale simulations.

... We find that D F 210 Fig. 4) which is comparable to estimates inferred for past earthquakes of similar magnitude (Viesca and Garagash, 2015). We note that fracture energy on both fault planes is roughly equivalent for both models in panels a and b, despite their distinct rupture dynamics (Guatteri and Spudich, 2000). ...

... The average fracture energy computed accounting only for fault cells with slip larger than 20% of average slip is 0.7M J/m 2 for Family (B) and 0.61M J/m 2 for Family (C). These averages are smaller than those obtained for models of Family (Hom) (Section 4.1) and consistent with proposed scaling laws between fracture energy and seismic moment (Viesca and Garagash, 2015;Tinti et al., 2005). Fig. 8 compares synthetic velocity waveforms, with selected observed data in the near-source region. ...

The 2016 Central Italy earthquake sequence is characterized by remarkable rupture complexity, including highly heterogeneous slip across multiple faults in an extensional tectonic regime. The dense coverage and high quality of geodetic and seismic data allow us to image intriguing details of the rupture kinematics of the largest earthquake of the sequence, the Mw 6.5 October 30th, 2016 Norcia earthquake, such as an energetically weak nucleation phase. Several kinematic models suggest multiple fault planes rupturing simultaneously, however, the mechanical viability of such models is not guaranteed.
Using 3D dynamic rupture and seismic wave propagation simulations accounting for two fault planes, we constrain “families” of spontaneous dynamic models informed by a high-resolution kinematic rupture model of the earthquake. These families differ in their parameterization of initial heterogeneous shear stress and strength in the framework of linear slip weakening friction.
First, we dynamically validate the kinematically inferred two-fault geometry and rake inferences with models based on only depth-dependent stress and constant friction coefficients. Then, more complex models with spatially heterogeneous dynamic parameters allow us to retrieve slip distributions similar to the target kinematic model and yield good agreement with seismic and geodetic observations. We discuss the consistency of the assumed constant or heterogeneous static and dynamic friction coefficients with mechanical properties of rocks at 3-10 km depth characterizing the Italian Central Apennines and their local geological and lithological implications. We suggest that suites of well-fitting dynamic rupture models belonging to the same family generally exist and can be derived by exploiting the trade-offs between dynamic parameters. Our approach will be applicable to validate the viability of kinematic models and classify spontaneous dynamic rupture scenarios that match seismic and geodetic observations as well as geological constraints.