Numerical study on multi-scaling earthquake rupture

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L266, doi:1.129/23gl1878, 24 Numerical study on multi-scaling earthquake rupture Hideo Aochi Institut de Radioprotection et de Sûreté Nucléaire, France Satoshi Ide Department of Earth and Planetary Science, University of Tokyo, Japan Received 2 September 23; revised 21 November 23; accepted 17 December 23; published 24 January 24. [1] A new numerical scheme using a renormalization and a 3D boundary integral equation method is proposed to simulate a multi-scaling dynamic rupture of earthquakes: How a small earthquake grows up to a large one in spatially heterogeneous field of critical slip-weakening distance D c (fracture energy G c )? We examine the case where D c grows according to a hypocentral distance L (D c / L b ). When b =1, we succeed to show numerically that a rupture propagates at a constant rupture speed in uniform initial stress field. This result still keeps the scaling relation of G c and D c inferred for earthquake size, however no scale-dependent initial process is required. The break of the proportional relation changes rupture speed as well as slip velocity to keep the energy balance. The rupture is accelerated up to a speed even faster than the shear wave velocity (b < 1) or naturally arrested (b > 1). INDEX TERMS: 721 Seismology: Earthquake parameters; 729 Seismology: Earthquake dynamics and mechanics; 726 Seismology: Theory and modeling; 81 Structural Geology: Fractures and faults. Citation: Aochi, H., and S. Ide (24), Numerical study on multi-scaling earthquake rupture, Geophys. Res. Lett., 31, L266, doi:1.129/ 23GL Introduction [2] When we discuss scaling problems in earthquake rupture, it is common to use fracture energy G c or critical slip displacement D c of the slip-weakening friction law, which is mathematically expressed in the form; tðu Þ ¼ t b ð1 u=d c ÞHð1 u=d c Þþt r : ð1þ Here H() is a Heaviside function and t b is the breakdown strength drop defined by the difference between peak and residual strengths, (t p t r ). The fracture energy follows the relation G c = 1 2 t b D c. Indeed, Ide and Takeo [1997] estimated G c of an order of 1 6 J/m 2 for earthquakes of magnitude around 7, while it is thought of an order of 1 4 J/m 2 in laboratory experiments [e.g., Scholz, 22]. It is also proposed that D c may reach in an order of 1 m for an earthquake of a magnitude 8, whereas it is actually observed in mm for rock experiments [e.g., Ohnaka, 23]. These D c -scaling and G c -scaling are consistent, as it is known in seismology that average stress change is about a few MPa independent of the scale. Shibazaki and Matsu ura [1992] modeled a concept of scale-dependent D c, leading to scale-dependent nucleation theory. Copyright 24 by the American Geophysical Union /4/23GL1878 [3] However, such parameters as D c and G c do not always need to keep a value of the same order in a fault system, because, in general, earthquakes repeatedly occur in a single fault system with a different magnitude of more than 6. This implies the existence of heterogeneity in D c or G c of an order of 1. This paper is concerned about this heterogeneity, which we shall think as an intrinsic feature on the fault to control important aspects of earthquake rupture. Although rupture behaviors based on energy balance at a crack tip have been studied analytically for mainly 2D models [e.g. Freund, 199], we need a flexible numerical scheme to solve more realistic problems. However, it has been very difficult to numerically do it. When we aim to model a large earthquake, it is customary to take a relatively larger model region, bigger grid size and longer time step than those for a small event. This is allowed because the scale-dependent character of D c or G c is explicitly or implicitly taken into account so that the relatively large grid and time step have resolution enough to follow the change in friction. A single grid for a large earthquake (a few hundred meters for a M7 event) is usually much larger than the rupture size of a small event (a hundred meter even for a M3 event). This is why no one can answer to the principal physical question how a large earthquake is different from small ones, or whether a small one grows up eventually to a large one [Ellsworth and Beroza, 199]. It has been impossible to figure out what really happens within a single grid of a large earthquake especially in its initial stage. This is due to the fact in how the grid size is chosen to study small and large events. Alternatively, we shall first propose a new numerical procedure to handle efficiently multi-scaling rupture events, and then consider the effect of heterogeneous, still scale-dependent distribution of D c or G c. 2. Numerical Procedure [4] The boundary integral equation method (BIEM) for a single planar fault in unbounded, homogeneous elastic medium [Fukuyama and Madariaga, 1998] is very useful for our purpose. The stress field t(~x, t) is generally written in the form of the spatio-temporal convolution (BIE) of the Green s function and the slip velocity history _u(~x, t)onthe fault plane. Namely the stress t ijk on the fault element can be rewritten in the discrete form of the summation of the discrete kernel P (i l)(j m)(k n) and discrete slip velocity V lmn, which is supposed to be constant on each spatial grid (s s) during each time step (t). Here the suffix (i, j) and (l, m) represent the spatial grid number, and k and n are for the time step, respectively. L266 1of

2 Figure 1. Schematic illustration of renormalization procedure in BIEM. Slip velocity on a single renormalized grid requires the 16 spatial grids over 4 time steps. [] Since the discrete slip velocity is a box function both in space and in time, it allows us to renormalize its grid size and time step, by conserving the rate of seismic moment (slip velocity grid surface duration). Letting a new grid size s, a new time step t and the slip velocity renormalized on the new grid V *, we will see V i j k * s2 t ¼ V ijk s 2 t ; ð2þ where the summation is taken over the grids covering the same fault area and over the same time duration. This procedure is also graphically shown in Figure 1. When we scale up by a factor of 4, for example, then s =4s, t = 4t, and V * i j k = (1/64)V ijk. [6] On the other hand, it is physically required that the total fracture energy on the fault is also conserved through this renormalization process. We can write it down in the same form as above; G i j c s 2 ¼ G ij c s2 ; ð3þ where G c * i j is the renormalized fracture energy on a grid (i, j ) based on those G ij c in the smaller scale. In the case where we renormalize by a factor of 4, G c * i j is thus composed by 16 grids in the smaller scale. As we assume that t b is constant for all scales, G c in the above equation can be replaced by D c. [7] By repeating this procedure, we are now able to follow the transit process from a very microscopic event to a macroscopic earthquake rupture. 3. Self-Similar Rupture Propagation [8] We shall see what happens in different scales during a single rupture process. In spite of the concept of scaledependent D c or G c explained above, some people consider that rupture always starts at a very small point corresponding to a very small D c or G c as observed in rock experiments. To avoid the contradiction between two opinions, D c or G c should grow up following the progress of rupture propagation itself. In fact this was implied by Richards [1976] in his review of analytic solution of 3D self similar rupture propagation. Such growth of G c,orr (resistance)-curve in the fracture mechanics, is widely observed in tensional fracture experiments [e.g., Lawn, 1993]. Here we generally suppose a simple relation with constant b; D c / L b ; where L is the hypocentral distance. There are a few explications for this general scaling law (Andrews, personal communication, 23). One is that D c is intrinsically distributed in the fractal manner on the fault. Rupture may starts almost everywhere at the smallest patch with the smallest G c (D c ), so that its averaged feature from the beginning point increases with distance to some extent, such a way as the above equation. Another is that D c is not a fixed property on fault, but arises as a result of dynamic process itself during the rupture. Size dependence of G c in material fracture arises because energy is lost to plastic yielding in a zone that becomes larger as the fracture length becomes larger. Andrews [1976] reported such the similar effect in dynamic shear rupture. [9] Table 1 shows model parameters for different scale in the case of b = 1, where D c is proportional to L. The other elastic parameters are common through the simulations; rigidity of 32.4 GPa and P/S wave velocity of 6./3.46 km/s. Strength t p, residual strength t r and initial stress t are MPa, MPa and 3 MPa, respectively, for all stage of simulations (scale independent). At the minimum scale stage, we suppose that a finite frictionless crack of a radius of 1. m suddenly appears at t =, so that rupture initiates outward spontaneously. We use the fixed grid size and time step in each stage. Table 1. Model parameters used in each stage of the simulation Simulation Stage 1st... Grid Size s m 1.6 m km Time Step t.33 ms.133 ms s Critical Slip Distance D c (L ).1 mm mm... m Reference Distance L 1. m 6. m km Magnitude at 24 t D c value is given at distance L as a reference following Equation (4). ð4þ 2of

3 slip velocity, m/s.4 slip, m shear stress, MPa Renormalization x 4 Figure 2. Snapshot of a single rupture propagation at different stages of simulation with different scales. As shown in detail at the seventh stage, slip velocity after the renormalization is very small at the beginning, however dynamic rupture propagation actually occurs in the smaller scales. Slip velocity in blue is the one renormalized from the earlier stage, whose model region is shown as a small frame. [1] We simulate over a model region of spatial grids up to 24 time steps, and then renormalize the result by a factor of 4, according to Equation (2), into the first 6 time steps of the next stage. Figures 2 shows snapshots of a single rupture propagation in different scales. After 6 times of renormalizations, we arrive in the seventh stage whose dimension is 496 (= 4 6 ) times larger than the smallest stage. We first notice that the rupture propagates in the same way at any stage with a constant rupture velocity. The spatial patterns and the value of slip velocity and shear stress do not change for any scale, while total slip actually increases. We need no spatial heterogeneity in initial stress field to generate self-similar rupture propagation. Although we have not yet constrained any condition to stop this rupture, our self-similar rupture propagation can be arrested at any size to lead a particular size of earthquake. In other words, all size of earthquake may start with the minimum size of crack. In this way, the condition D c / L produces the self-similar rupture propagation. This numerical result is also consistent with theoretical implication [e.g., Richards, 1976]. We suppose a significantly small frictionless patch to nucleate this spontaneous rupture propagation, because we do not introduce such heterogeneity in strength as observed in the experiments [e.g., Ohnaka et al., 1987]. The self- Shear S tress [MPa] t= 6 t= 6 (28,32) (3,32) (x 1, x 2) = (32,32) t= 6 t= 6 t= 6 (28,3) (3,3) (32,3) t= 6 t= 6 (28,28) t= 6 (3,28) t= 6 (32,28) Normalized Fault Slip Figure 3. Comparison of stress-slip constitutive curves before and after renormalization at the stage. Grid position (32, 32) is the closest to the center of rupture. Bold line represents the stress-slip curve calculated by BIE, and broken bold line is for the constitutive law (1). Each of the circles shows the position at time step and 6, respectively. Spontaneous process after time step 6 leads to that two lines are identical. Grey thin lines are stress-slip curve on corresponding smaller grids before renormalization. 3of

4 1. Normalized Seismic Moment Release st 3rd Time Step Figure 4. Evolution of seismic moment release function. The lines except for the first and second stages are identical. similar rupture propagation shown here implies that any particular size of earthquake needs no scale-dependent nucleation process and that all may begin from the smallest patch. [11] We have to check the stability of this renormalization process. In order to follow correctly rupture progress, it is generally required that the cohesive zone behind the rupture front is expressed by plural grids, and that slip-weakening process (u < D c ) is also followed by plural time steps. When we carry out the renormalization every 16 time steps (instead of 24), which corresponds the first 4 time steps (instead of 6) in the next stage, we get the same results, too. But if we take much shorter time step for this renormalization, the rupture cannot continue correctly. As shown in the snapshot of the seventh stage of Figure 2, we cannot distinguish a rupture front clearly at its early stage (until 3 time steps), nor even any clear evidence of slip at the beginning, although the dynamic rupture actually occurs in the smaller scale. [12] We examine the consistency between V and fracture energy G c, as we renormalized them. The stress can be independently recalculated by either of the constitutive law (1) or of the BIE. Figure 3 shows the comparison among two descriptions of stress evolution after the renormalization and the relations corresponding to the same grid area before the renormalization. As grid is closer to the hypocenter, stress accumulation is apparently lower than the actually required peak strength t p, so that slip weakening rate t b /D c becomes less steep and apparent D c becomes longer. This is because the renormalization procedure has a Time Step Dc, cm S P Hypocentral Distance Figure. Comparison of rupture propagation for different b. The curves represent the evolution of rupture front in the inplane direction. Both axes are normalized by the minimum sizes of grid size and time step. D c is also shown by side. Dotted lines correspond to the distance L b. 4of

5 smoothing effect. However this discrepancy is localized just inside of the area that already ruptured, especially at the beginning. Furthermore, the fracture energy actually consumed is in total conserved with respect to the required one. Therefore we can conclude that this smoothing effect does not influence the later stages after the renormalization. [13] Finally we show the evolution of seismic moment release function, normalized at each stage, in Figure 4. Since we give a finite frictionless crack at t =, we observe somehow its influence at the beginning. However, the relation curve for the second stage starts with an infinitesimal value, and the lines are identical after the third stage. This also infers that rupture starts with a point for any scale and propagates outward with a constant speed, keeping the self-similarity. 4. Break in Self-Similarity [14] Next we consider the case of b 6¼ 1. As shown in Figure, we keep a linear relation (b = 1) for hypocentral distance shorter than L b (48 times of the smallest grid size, corresponding to 192 m from Table 1), but suppose different b (b = 1.1,.8 and., for example) for a longer distance than L b. Figure shows the evolution of rupture front in the in-plane direction. We plot the simulation results until the 6th stage, so that the smallest scale is invisible and discrete steps appeared in larger scales correspond to one spatial grid size at each stage. We can also confirm the self-similar rupture propagation at a constant rupture velocity for the case b = 1, seen in Figure 2. [1] In the case of b = 1.1, rupture propagation is decelerated gradually according to the energy balance around the rupture front, and finally ends up a magnitude 4.2 event. This explains how the earthquake size is determined in the term of fracture energy balance. The break toward b > 1 naturally arrests the rupture propagation. On the other hand, the rupture is accelerated for b < 1. We note that slip velocity also increases according to the rupture progress so as to keep energy balance totally around the rupture front. The case of b =. has an extreme feature that rupture velocity exceeds even the shear wave velocity. This is the situation usually treated in most rupture simulations with a fixed grid size and a scale-invariable D c. Super shear rupture is rather rare in experiments and field observations. This implies that the spatial heterogeneity of fracture energy in the real field is close to the case b = 1, and tends to break toward b > 1 in some scale for the arrest of rupture.. Summary [16] We developed a numerical scheme combining a renormalization process with the boundary integral equation method, in order to accomplish a single multi-scaling earthquake rupture process in the same framework. This allows us to follow how a small earthquake grows up to a large one by a several order of its scale. We showed the self-similar rupture propagation in the case where the critical slip-weakening distance D c is proportional to the hypocentral distance L. For this case, we did not need any heterogeneity in the initial stress. We confirmed numerically that a small event becomes larger as a cascading rupture style without preparing any scale-dependent nucleation process. Then we showed that the break of the proportional relation toward b > 1 naturally arrests the rupture, so that we can generate any size of earthquake in this framework. For the case of b < 1, we also observed no-self-similar behavior in rupture process, where the rupture accelerates even at a rupture velocity higher than the shear wave velocity. [17] Acknowledgments. We thank D. J. Andrews and an anonymous reviewer for their valuable comments. This work is contribution to the DaiDaiToku Project, MEXT, Japan. References Andrews, D. J. (1976), Rupture propagation with finite stress in antiplane strain, J. Geophys. Res., 81, Ellsworth, W. L., and G. C. Beroza (199), Seismic evidence for an earthquake nucleation phase, Science, 268, Freund, L. B. (199), Dynamic fracture mechanics, Cambridge Univ. Press, Cambridge, UK. Fukuyama, E., and R. Madariaga (1998), Rupture dynamics of a planar fault in a 3D elastic medium: Rate- and Slip-weakening friction, Bull. Seismol. Soc. Am., 88, Ide, S., and M. Takeo (1997), Determination of constitutive relations of fault slip based on seismic wave analysis, J. Geophys. Res., 12(B12), Lawn, B. (1993), Fracture of brittle solids, edition, Cambridge Univ. Press, Cambridge, UK. Ohnaka, M. (23), A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture, J. Geophys. Res., 18(B2), 28, doi:1.129/2jb123. Ohnaka, M., Y. Kuwahara, and K. Yamamoto (1987), Constitutive relations between dynamic physical parameters near a tip of the propagating slip zone during stick-slip shear failure, Tectonophys., 144, Richards, P.G. (1976), Dynamic motions near an earthquake fault: A threedimensional solution, Bull. Seismol. Soc. Am., 66, Scholz, C. H. (22), The mechanics of earthquake and faulting, edn., Cambridge Univ. Press, Cambridge, UK. Shibazaki, B., and M. Matsu ura (1992), Spontaneous processes for nucleation, dynamic propagation, and stop of earthquake rupture, Geophys. Res. Lett., 19(12), H. Aochi, Institut de Radioprotection et de Sûreté Nucléaire, IRSN/DEI/ SARG/BERSSIN, BP17, Fontenay-aux-Roses, 92262, France. (hideo. aochi@irsn.fr) S. Ide, Department of Earth and Planetary Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, , Japan. (ide@eps.s.u-tokyo. ac.jp) of