Rate-and-state friction: From Analysis to Simulation E. Pipping, R. - PowerPoint PPT Presentation

Thrust faults Strike-slip faults References Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber 1 , M. Rosenau 2 , O. Oncken 2 , A. Mielke 3 1 Mathematisches Institut, Freie Universität Berlin, 2 Geologische Systeme: Lithosphärendynamik, GeoForschungsZentrum Potsdam 3 Weierstraß-Institut für Angewandte Analysis und Stochastik March 28, 2017 Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Types of faults: two examples (a) A thrust fault 1 (b) A strike-slip fault: Vertical, asymmetric arrangement Horizontal, symmetric arrangement 1 Also known as a reverse dip-slip fault (as opposed to a normal fault). Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Outline 1 Thrust faults Friction frameworks Continuum-mechanical model 2D simulation (in detail) 3D simulation (at a glance) 2 Strike-slip faults Modelling attempt, open questions Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Prime example: subduction zone Volcanic Arc Trench Rigid Backstop Seismogenic Zone Continental Aseismic Lithosphere Continental Asthenosphere Seismic Aseismic Figure: A subduction zone: the source of megathrust earthquakes Modelling situation: bilateral contact; friction. Simplifications: small deformation; small strain; one-body problem (bilateral contact with half-space); linear Kelvin–Voigt viscoelasticity. Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Friction frameworks Rate-and-state friction Figure: Velocity-stepping test; | σ t | = µ | σ n | + C , σ n = const . Measurements/ageing/slip law Westerly granite inside a double direct shear apparatus Source: M. F. Linker and J. H. Dieterich. “Effects of Variable Normal Stress on Rock Friction: Observations and Constitutive Equations”. In: Journal of Geophysical Research: Solid Earth 97.B4 (1992), pp. 4923–4940. doi : 10.1029/92JB00017 Clearly, we can write µ ( t ) = µ ( V )( t ) but not µ ( t ) = µ ( V ( t )). Ruina’s model takes the form ˙ µ ( t ) = µ ( V ( t ) , θ ( t )) and θ ( t ) = g ( θ ( t ) , V ( t )) Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Friction frameworks Restricted rate-and-state friction We consider here only the case µ ( t ) = µ ( V ( t ) , α ( t )) with α ( t ) + A ( α ( t )) = f ( V ( t )). ˙ with a monotone operator A and Lipschitz-continuous f . Example: Dieterich’s ageing law ˙ θ = 1 − θ V L can be transformed to read α − e − α = − V ˙ θ 0 L with α = log( θ/θ 0 ). Not an example: Ruina’s slip law ˙ θ = − θ V L log θ V L does not fit into this framework. More on this matter in the appendix. Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Continuum-mechanical model Strong-strong formulation Our initial-value problem reads: find a displacement field u on Ω and a scalar state field α on the boundary segment Γ C such that σ = A ε ( ˙ u ) + B ε ( u ) in Ω × [0 , T ] ∇ · σ + b = ρ ¨ u in Ω × [0 , T ] u = 0 ˙ on Γ D × [0 , T ] σ n = 0 on Γ N × [0 , T ] ˙ u · n = 0 on Γ C × [0 , T ] − σ t = µ ( | ˙ u | , α ) | ¯ σ n | + C  ˙ for ˙ u � = 0 u  | ˙ u | on Γ C × [0 , T ] | σ t | ≤ µ (0 , α ) + C for ˙ u = 0  α + A ( α ) = f ( | ˙ ˙ u | ) on Γ C × [0 , T ] Note that we replace the (unknown) σ n with a fixed ¯ σ n . We also prescribe initial conditions on u , ˙ u , ¨ u , and α . Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Continuum-mechanical model Weak-strong formulation In a standard fashion we arrive at the weak formulation 2 b ( t ) ∈ ρ ¨ u ( t ) + A ˙ u ( t ) + B u ( t ) + γ ∗ ∂ Φ α ( t , · )( γ ˙ u ( t )) with A , B given by � � A v = � A ε ( v ) , ε ( · ) � and B v = � B ε ( v ) , ε ( · ) � . Ω Ω as well as the friction nonlinearities � Φ α ( t , v ) = ϕ α ( t , x , | v ( x ) | ) d x Γ C � v ϕ α ( t , x , v ) = µ ( r , α ( t , x )) | ¯ σ n | + C d r . 0 We interpret A as a superposition operator in α ( t ) + A ( α ( t )) = f ( | γ ˙ ˙ u ( t ) | ). 2 The precise solution spaces are mentioned on the next slide. Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Continuum-mechanical model Analytical results For the restricted type of rate-and-state friction (which makes additional assumptions on µ ), we have been able to show: • (2017) For any T > 0, we have a unique solution to the coupled weak-strong problem with u ∈ L 2 (0 , T , V ) , u ∈ L 2 (0 , T , V ) , u ∈ L 2 (0 , T , V ∗ ) ˙ ¨ α ∈ C (0 , T , L 2 (Γ C )) where V = { v ∈ H 1 (Ω) d : v = 0 on Γ D , v · n = 0 on Γ C } . • (2014) For certain time-discretisation schemes (e.g. Newmark, backward Euler), one needs to solve problems of the form � λ M � τ ρ + A + τ b n ∈ u n ( t ) + γ ∗ ∂ Φ α, n ( γ ˙ ˙ u n ) B λ B If each step is no larger in size than a certain constant, then all time u n ∈ V and α n ∈ L 2 (Γ C ). steps have unique solutions u n , ˙ u n , ¨ In both cases, a fixed-point map is employed that turns into a contraction for sufficiently small time increments. The 2014 result thus also shows that a fixed-point iteration will converge regardless of the starting point. Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References 2D simulation (in detail) Video We run a simulation with the dimensions of (and parameters taken from) a lab-scale analogue model (more on that later). Figure: A still frame from the video Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References 2D simulation (in detail) Spatial resolution Figure: Actual spatial resolution of the simulation (wireframe / vertices as dots) Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References 2D simulation (in detail) Surface uplift, numerical performance · 10 − 6 20 995 10 surface uplift [ m ] 5 time [s] 2 . 5 0 990 − 2 . 5 − 5 − 10 985 − 20 0 . 00 0 . 20 0 . 40 0 . 60 0 . 80 6 4 2 15 10 5 10 − 1 10 − 2 10 − 3 distance from trench [m] TS size [s] FP iter. MG iter. Figure: Surface uplift, performance of the numerical components: (1) Adaptive time-stepping, (2) Fixed-point iteration, (3) TNNMG 3 3 Truncated Nonsmooth Newton Multigrid Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References 2D simulation (in detail) Comparison: laboratory and experiment Our simulation is based on the analogue model first presented here: M. Rosenau, R. Nerlich, S. Brune, and O. Oncken. “Experimental insights into the scaling and variability of local tsunamis triggered by giant subduction megathrust earthquakes”. In: Journal of Geophysical Research: Solid Earth 115.B9 (2010). doi : 10. 1029/ 2009JB007100 This allows us to compare our numerical results with lab measurements. simulation experiment 5 10 20 40 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 03 0 . 06 0 . 12 recurrence time [s] rupture width [m] peak slip [mm] Figure: We isolate three key quantities. Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References 3D simulation (at a glance) Figure: Computational 3D grid slip rate [m / s] 0 . 30 1 × 10 − 3 0 . 20 3 × 10 − 4 depth [m] 0 . 10 1 × 10 − 4 0 . 00 3 × 10 − 5 − 0 . 10 1 × 10 − 5 − 0 . 20 3 × 10 − 6 − 0 . 30 1 × 10 − 6 0 . 2 0 . 4 0 . 2 0 . 4 0 . 2 0 . 4 0 . 2 0 . 4 0 . 2 0 . 4 0 . 2 0 . 4 t 0 ≈ 991 s t 0 + 0 . 14 s t 0 + 0 . 21 s t 0 + 0 . 27 s t 0 + 0 . 34 s t 0 + 0 . 46 s distance from trench [m] Figure: Coseismic evolution of sliding rate contours along seismogenic zone Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke

Thrust faults Strike-slip faults References Modelling attempt, open questions Reminder: one-body problem on thrust fault Strong-strong problem: u on Ω, α on Γ C . σ = A ε ( ˙ u ) + B ε ( u ) in Ω ∇ · σ + b = ρ ¨ in Ω u u = 0 ˙ on Γ D σ n = 0 on Γ N u · n = 0 ˙ on Γ C − σ t ∈ ∂ϕ α ( t , · )( | ˙ u | ) on Γ C α + A ( α ) = f ( | ˙ ˙ u | ) on Γ C (a) Thrust fault (b) Strike-slip fault Rate-and-state friction: From Analysis to Simulation E. Pipping, R. Kornhuber, M. Rosenau, O. Oncken, A. Mielke