Effect of normal stress during rupture propagation along nonplanar faults

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B2, 2038, /2001JB000500, 2002 Effect of normal stress during rupture propagation along nonplanar faults Hideo Aochi and Raúl Madariaga Laboratoire de Géologie, École Normale Supérieure, Paris, France Eiichi Fukuyama National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Japan Received 28 February 2001; revised 17 August 2001; accepted 22 August 2001; published 22 February [1] We study a symmetrical three-forked shear fault under simple triaxial stress. Rupture initiates in the central primary plane and propagates toward the branching point. One of the two branches is in the compressional quadrant, while the other is in the tensional domain. We study the following question: Does rupture always prefer propagating along the tensional branch because of the effect of normal stress? When the primary plane is located in the most favorable direction as determined by Mohr-Coulomb criterion, one finds that the compressional branch is almost always preferred for various dynamic coefficients and confining pressures. Simultaneous rupture propagation on both branches appears only under a very special condition found by Aochi et al. [2000b]. These results indicate that shear stress changes produced by the rupture front dominate during dynamic rupture propagation, compared to normal stress changes. This result depends completely on the orientation of the whole fault system. In all cases, we can explain the branch selection by analyzing the applied initial shear stress and the fault properties. Change of normal stress may play a role similar to shear stress change for disjoint faults since the effect of normal stress change sometimes dominates. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 7260 Seismology: Theory and modeling; KEYWORDS: Spontaneous rupture propagation, boundary integral equation method, branched fault, absolute stress level, Coulomb law 1. Introduction [2] There is a broad consensus that fault geometry is very important in determining the rupture process of earthquakes [King and Nábělek, 1985; Sibson, 1986]. This process has been studied numerically and theoretically by Nielsen and Knopoff [1998] as a static problem and by Harris and Day [1993, 1999], Kase and Kuge [1998], and others as a dynamic problem. Aochi and Fukuyama [2002] showed that fault branching or branch selection of rupture propagation was also an important problem for the 1992 Landers, California, earthquake. [3] Tada and Yamashita [1997] simulated kinematic rupture propagation along a two-dimensional (2-D) prebranched fault system and reported that energy dissipation on each branch facilitated the arrest of rupture propagation. Kame and Yamashita [1999] and Seelig and Gross [1999] simulated spontaneous crack growth in a 2-D intact material and allowed crack branching. More recently, Aochi et al. [2000b] numerically modeled spontaneous rupture propagation along a preexisting branched fault system in a 3-D medium, and they reported that branch selection by rupture was very sensitive to the applied initial shear stress. For the purpose of general understanding of dynamic rupture under complex fault geometry, Aochi et al. [2000b] used a simple slipweakening law depending only on shear stress. Although they discussed what may happen with a Coulomb-type friction law based on their results, it is still necessary to clarify the effect of normal stress on rupture propagation. [4] Harris et al. [1991] and Harris and Day [1993, 1999] numerically modeled dynamic rupture propagation on parallel offset faults. Their results indicated that ruptures could more easily Copyright 2002 by the American Geophysical Union /02/2001JB000500$09.00 jump to a parallel fault located in a region of tensional change of normal stress than to one in the compressional range. However, their simulation method, finite differences (FDM), is restricted to parallel fault geometries because of its spatial grid partition. In the present paper, we use a boundary integral equation method (BIEM), which is more flexible for modeling complex fault geometries, and investigate more general branching. We simulate dynamic rupture propagation along nonplanar faults with a slipweakening Coulomb s friction law and investigate the effect of normal stress through a parametric study. In general, the increase of normal stress applied on the fault disturbs rupture, and the decrease of normal stress enhances rupture initiation because according to Coulomb friction, peak strength (yielding stress) is proportional to normal force. The question that we ask is Does a rupture always turn toward the tensional direction? 2. Model 2.1. Geometrical Setting and Numerical Method [5] We consider a symmetrical three-forked branched fault which consists of three planar faults as shown in Figure 1. The fault branches are symmetric with respect to the primary plane on which the rupture is assumed to initiate. For a right-lateral strikeslip fault, compressional stress must increase on the upper branch in Figure 1 and decrease on the lower branch, whereas shear stress changes on both branches in the same way. We will designate them as the compressional and tensional branches, respectively. We take the x 1 axis along the fault, the x 2 axis perpendicular to it, and the x 3 always normal to the fault plane. As shown in Figure 1, we represent the branching angle measured counterclockwise from the primary plane as a. [6] For this kind of nonplanar fault model the 3-D dynamic boundary integral equation method (BIEM) is accurate and con- ESE 5-1

2 ESE 5-2 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS Figure 1. Illustration of a three-forked symmetrical branched fault. We refer to each subfault as primary plane, compressional branch, and tensional branch, respectively. Numbers along the faults represent grid positions. We take a local coordinate x 1 along the fault direction, and x 2 perpendicular to it on the plane. venient [Aochi et al., 2000a]. We can estimate shear and normal stress at each point on the fault and at each time step. A series of 3-D BIEM proposed by Cochard and Madariaga [1994], Fukuyama and Madariaga [1995, 1998], Tada et al. [2000], and Aochi et al. [2000a] have the advantage that the instantaneous term is isolated from the past history for the calculation of the BIEM in the time-space domain. Applying a discretization where a constant slip velocity (V t ) is assumed within each spatial grid (s s) during each time step (t), we can briefly write the BIE in the following symbolic notation: t ijk 3s ¼ X3 t¼1 " # P3s=t 0 V ijk t þ Xk 1 X P ijk:lmn 3s=t Vt lmn ; ð1þ n¼0 l; m where s and t represent the components in local coordinates, (i, j) and (l, m) represent the discretized position on the fault on the (x 1, x 2 ) plane, and k and n represent the discretized time step. The 0 first term on the right-hand side P 3s/t ( P ijk:ijk 3s/t ) is the instantaneous stress term [Fukuyama and Madariaga, 1995, 1998] and represents the instantaneous contribution of the current slip velocity to the stress at the same position. The second term in ijk:lmn (1) contains the contribution of the past slip rate history. P rs/t indicates the stress kernel in rs component at (i, j, k) due to the t direction unit slip at (l, m, n). Figure 2 shows the summation region, which corresponds to the convolution region which satisfies causality in the BIEs. From (1) and Figure 2, we i remark that slip on any of the other grids V 0 j 0k t,(i 0, j 0 ) 6¼ (i, j), does not affect stresses at the reference point (i, j) during the current time step k, when we take the time step t less than the travel time of P wave for half a grid size. We take t = s/(2v p ), so that the solution of the BIEM (equation (1)) is stable, as proposed by Fukuyama and Madariaga [1998] for 3-D simulations. [7] Let us remark that as long as we consider shear rupture without any fault-opening displacement mode (V 3 0), the instantaneous term due to shear slip displacement on the fault affects only the shear stress components, t 31 and t 32. That is, for 0 example, there exists only P 31/1 for V 1. On the other hand, normal stress t 33 depends only on the past slip history [Aochi et al., 2000a; Tada et al., 2000]. Finally, as can be deduced from symmetry ijk:lmn arguments, the contribution of shear slip to normal stress P 33/1 is always zero when the source and receiver are on the same plane Slip-Weakening Coulomb Friction Law [8] We introduce Coulomb s friction law, which has been widely used to study dynamic friction on faults [e.g., Dieterich, 1979; Ruina, 1983]. The peak strength t p (yield stress) and residual stress level t r (dynamic stress level) are given by the static and dynamic frictional coefficients, m s and m d, multiplied with normal stress s n, respectively: t p ¼ m s s n ð2þ t r ¼ m d s n ð3þ The transition process from the peak strength t p to the residual stress level t r during the initial stages of dynamic rupture is generally expressed by slip-weakening law originally proposed by Ida [1972] and Palmer and Rice [1973]: sðuþ ¼ t r þ t b ð1 u=d c ÞHD ð c uþ; ð4þ where t b ¼ t p t r ¼ s n ðm s m d Þ ð5þ u is fault slip, and H( ) is the Heaviside function. D c and t b are the critical slip-weakening displacement and breakdown strength Figure 2. Schematic diagram of the summation area in the boundary integral equation (BIE). The stress at the reference point (i, j) and the time k is contributed by its own grid V ijk and the past slip history included under the wave cone of the P wave, the grids illustrated as dark shaded boxes. The outside of the wave cone as drawn with light shaded boxes, including the neighborhood grid V (i +1)jk, does not affect it.

3 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS ESE 5-3 Figure 3. Illustration of the stress field assumed in the present simulation. Parameters surrounded by a solid square are assumed a priori and others surrounded by a dashed square are evaluated using the assumed parameters as explained in the text. (a) External compression field around a symmetrical three-forked branched fault and (b) simulation parameters determined from the Mohr-Coulomb criterion. They are in the case of P = 100 MPa, m s = 0.6, t b = 10 MPa, t e = 5 MPa. drop, respectively. The shear strength of the fault s decreases linearly with ongoing slip u and is also an explicit function of normal stress s n in (3). More complex slip- or ratedependent frictions can be studied, but we do not think that they will change our main conclusions [see, e.g., Bizzari et al., 2001]. [9] Let us consider an external triaxial compression field (s 1 > s 2 > s 3 ), taking compression as positive, as illustrated at the left of Figure 3. For simplicity, we suppose that the fault is parallel to s 2 and that branching occurs on the plane of the maximum and minimum principal stress (s 1 and s 3 ). Rupture initiates on the primary fault that makes an angle 0 with the maximum principal stress axis s 1. This kind of situation corresponds to strike-slip faults with dip angle of 90, where the x 1 axis is the initial direction of fault strike. [10] The purpose of this paper is to investigate the effect of normal stress s n and of fault geometry represented by the branching angle a. Although there are many degrees of freedom in this problem, we assumed that the same external stress (s 1, s 2, s 3 ) is applied on both primary and branched faults. We also assume that the stress increase t e ( t p t 0 ), sometimes referred as the stress excess, and the breakdown strength drop t b are constant on the primary fault, where t 0 is the initial shear stress applied by external stresses s 1 and s 3. On the basis of this assumption we introduce the Mohr-Coulomb fracture criterion for determining the direction of primary fault 0 and other parameters. [11] We show this procedure graphically in Figure 3b. Here the input parameters assumed a priori are m s, the confining pressure P, a, t e, and t b. As a result, we will get m d, s 1, s 3, 0, and then the initial stress field all over the fault system. As Figure 3 shows, P and s satisfy and t 0 and s n 0 are initial shear and normal stresses applied on the fault. From Figure 3 we can write (6) as s 0 n t 0 ¼ s sin 2 ¼ P s cos 2 ð9þ for a fault plane inclined of a angle with respect to the maximum principal stress axis s 1. Assuming that the initial rupture plane coincides with the most likely direction expected from the Mohr circle for initial stress level, we get the angle of primary fault plane 0 as 0 ¼ p tan 1 m s From (2), (8), (9), and the assumed value of t e we get s ¼ and from (3) and t b, m s P t e m s cos 2 0 þ sin 2 0 ; ð8þ ð10þ ð11þ m d ¼ s sin 2 0 ðt b t e Þ : ð12þ P s cos 2 0 Finally, we express the external stresses as s 1 ¼ P þ s s 3 ¼ P s ð13þ where t 2 0 þ s0 n P 2¼ s 2 ; ð6þ P ¼ s 1 þ s 3 2 ; s ¼ s 1 s 3 ; ð7þ 2 and compute the initial stresses t 0 and s n 0 from (8) and (9). Note that the breakdown strength drop t b and the stress increase t e change as a function of the fault orientation. [12] Figure 3 shows the case in which we assume P = 100 MPa and m s = 0.6 globaly and t b = 10 MPa and t e = 5 MPa on the primary plane. The other parameters are determined as m d = 0.468, 0 = 29.5, s 1 = MPa, and s 3 = 52.9 MPa.

4 ESE 5-4 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS Figure 4. Dependency of the potential angle upon the confining pressure P for three static friction coefficients m s = 0.6, 0.3, and 0.12, respectively. Angles are measured counterclockwise from the direction of the primary fault plane 0, so that the compressional and tensional branches correspond to positive and negative angles, respectively. The region between two curves contains the angles such that initial shear stress t 0 is higher than residual stress level t r. Solid circles represent the conditions on the two branches (inclined angle of ±15 ) simulated in this study Dependency of Parameters [13] Before simulating dynamic rupture propagation we study the possibility that rupture propagates on the branched faults, based on the discussion by Aochi et al. [2000a]. We can determine the range of angles for which branching is permitted from the static Mohr circle illustrated in Figure 3. A branch can break only if the initial shear stress on it, t 0, is higher than the residual stress level t r. The angles are then determined by the intersection of the Mohr circle with line m d in Figure 3b. For example, the range of possible branching angles for the parameters adopted in Figure 3b is 19.5 < <45.5, which corresponds to 10.1 < a < Hereinafter we refer to this range of the angles as the potential branching angle. The potential angles should be mainly affected by the confining pressure P and the static frictional coefficient m s from the discussion in section 2.2. Figure 4 shows their dependency for the cases when m s = 0.6, 0.3, and 0.12, respectively. The region between two curves defines the potential area where the initial shear stress t 0 is larger than the residual stress level t r, so that spontaneous rupture is permitted. In the case of a high frictional coefficient (m d =0.6)the area is very narrow so that the range of fault orientations permitting rupture propagation is severely limited, while it expands for low frictional coefficient and for low confining pressure. [14] Figure 4 is a static study of the possibility of branching along inclined faults. Along a bending fault the criterion whether rupture can spontaneously propagate beyond the bending point is this angle [Aochi et al., 2000a], although Aochi et al. used a slipweakening law independent of normal stress. In dynamic rupture, stress transfer between the branches affects rupture even more, as reported by Aochi et al. [2000b]. Rupture sometimes propagated only on one of the branches regardless of the favorable initial condition on the other, and simultaneous rupture propagation on both branches required very delicate conditions. In section 3 we investigate the effect of normal and shear stress transfer through parametric studies and discuss their significance during dynamic rupture propagation. All the parameters used in these studies are summarized in Table 1. We take one typical parameter set of confining pressure P = 100 MPa and branching angle a =15 shown as circles in Figure 4. This point is located at the edge of the potential angle given for m s = 0.6, slightly inside their range for m s = 0.3 and well inside of it for m s = Simulation 1: Fault Orientation Favored by Static Stress [15] We show three typical situations observed in different simulations in Figure 5. The branching angles a are supposed to be +15 (compressional branch) and 15 (tensional branch). The static frictional coefficient m s is different in each of the rows of the Table 1. Parameters Used in This Simulation Parameter Value Unit Rigidity m 30 GPa P wave velocity V p 5.77 km/s S wave velocity V s 3.33 km/s Density r 2700 kg/m 3 Grid size s 400 m Time step ts/(2v p ) 0.03 s Breakdown strength drop a t b 10 MPa Stress increase a t e 5 MPa Critical slip displacement D c 0.8 m Static frictional coefficient m d 0.6/0.3/0.12 Confining pressure P 100 MPa Branching angle a ±15 a Defined on the primary fault plane. Parameters, such as dynamic frictional coefficient m d and the direction of primary fault 0 and 0, are determined based on the above parameters, as explained in text.

5 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS ESE 5-5 Figure 5. Snapshots of rupture propagation. Common parameters P = 100 MPa and a = ±15 are used. Static frictional coefficient m s is (a) 0.6, (b) 0.3, and (c) 0.12, respectively. Along the x 1 (horizontal) axis the primary plane is located from 30 to 10, the compressional branch from 11 to 30, and the tensional branch from 31 to 50 as shown in Figure 1. Figure 6. History of (left) shear stress and (right) normal stress for (a) m s = 0.6, (b) 0.3, and (c) 0.12, corresponding to Figures 5a, 5b, and 5c respectively. Location (5, 0) is shown as open circles, (13, 0) as diamonds, and (33, 0) as triangles, in x 1 x 2 coordinate, as also plotted in Figure 5.

6 ESE 5-6 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS Figure 7. Phase diagram of the change in normal stress and shear stress in the case of static frictional coefficient, curve a for m s = 0.6, curve b for 0.3, and curve c for They correspond to each case in Figures 5 and 6. Symbols show the initial stress state at the position shown in Figure 5. Figure 5. Rupture propagates in the same way until it reaches the branching point. In the case of m s = 0.6 (Figure 5a), rupture does not propagate along the branches. As m s becomes small, it can progress along the compressional branch as shown in Figure 5band finally along both branches in Figure 5c. We did not observe rupture propagation only along the tensional branch. [16] In order to understand these examples we studied the change of stress at particular points on the fault system. Figure 6 shows the history of shear and normal stress at three different points, and Figure 7 shows the corresponding phase diagrams, the stress trajectory in the shear normal stress plane. Points (x 1, x 2 )= (5s, 0), (13s, 0), and (33s, 0) are located on the primary plane, the compressional branch, and the tensional branch, respectively. The history of stress at (5s, 0) is the same for all cases until rupture reaches the branching point, except for a constant baseline. In the case where rupture did not propagate on the branches, shown in Figure 5a, normal stress due to rupture propagation on the primary plane increases in the compressional branch and decreases in the tensional branch, whereas shear stress increases on both branches. This relation is the same as that expected from the external loading system. On the other hand, in the case where rupture propagates along the compressional branch as in Figure 5b, normal stress actually increases on the tensional branch due to rupture propagation along the compressional branch, whereas shear stress does not reach the failure point but decreases. We remark again that rupture does not cause changes of normal stress on the same plane. Actually, it shows the reverse tendency for the case in which rupture progressed along both branches, case in Figures 5c and 6 and curve c in Figure 7. In all cases, we observed that the change of normal stress is small compared to that of shear stress, especially near the rupture front. [17] Although these results may seem counterintuitive, they can be understood from Figures 4 and 7. In the case of m s = 0.6 the branch inclined 15 is located near the boundary of the potential fracture angle area, while it is completely inside the range of possible branching angles for m s = The former situation means that initial stress, t 0, on both branches is almost at the same level as residual stress, t r, and spontaneous rupture propagation is very difficult. On the other hand, for m s = 0.12, initial shear stress t 0 is much larger than the residual stress t r. Aochi et al. [2000b] already reported that simultaneous rupture propagation along both branches requires a very delicate condition without consideration of normal stresses. In our simulation, rupture propagated only along one branch in the case of m s = 0.3 (Figure 5b), although the branching angle a is within the potential range (21.8, 16.0 ). [18] We investigate now the reason why rupture preferred the compressional branch. Figure 7 shows the difference in stress evolution in each fault branch. It is clear from Figure 7 that the change of shear stress due to the passage of the rupture front is dominant. From geometrical considerations, stress increase for fracture (distance to the line of static friction m s ) is the same for both branches. However, possible stress drop (distance to the line of dynamic friction) is always larger on the compressional branch than on the tensional one, as shown in detail in Table 2. In other words, slip-weakening rate ( t b /D c ) is greater on the compressional branch. That is why the rupture propagates preferentially on the compressional branch in the case of intermediate values of the frictional coefficient (m s = 0.3). These results can be interpreted together with the results by Aochi et al. [2000b], who introduced a simple slip-weakening law independent of normal stress. Rupture propagation can be determined by the fault parameters t p and t r and the applied shear stress t 0. This implies that discussing the direction of rupture propagation on a branched fault system in terms of shear stress is generally sufficient. 4. Simulation 2: Fault Orientation Preferred by Dynamic Stress [19] We observed that rupture preferred the compressional branch to the tensional one when it propagated along one of the branches and that this could be explained by the relation between the initial shear stress and fault parameters. It is clear that this situation, described in Figure 3, may easily change with fault orientation. As another possibility of rupture initiation, we assume that the ideal state inferred from the Mohr circle is obtained after fracturing. That is, as shown in Figure 8b, we define the direction of primary plane for the dynamic frictional coefficient m d instead of Table 2. Comparison of the Initial Condition on Each Fault Between Two Different Situations (Simulations 1 and 2) for the Case of m s =0.3 a Simulation 1 Simulation 2 Tension Primary Compression Tension Primary Compression 0 Initial normal stress s n Initial shear stress t Peak strength t p Residual stress t r Stress increase t e Breakdown strength drop t b a Each situation is explained in Figures 8a and 8b. Corresponding simulations results are shown in Figures 5b and 9, and then each phase diagram of stress is shown in Figures 7 and 10, respectively. Units are MPa.

7 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS ESE 5-7 Figure 8. Mohr circle for two simulations. (a) Stress field defined by 0 as used in simulation 1 and (b) stress field defined by 0 as in simulation 2. The primary plane 0 is directed toward the direction where the initial stress exceeds the residual stress in the highest. Both cases correspond to m d = 0.3. Small circles show the initial stress state on the primary fault plane, and triangles and squares show it on each branching fault (a = ±15 ). See also Table 2. the static friction m s. Hereinafter we call this angle 0, which we write from (10) as and from (11) as so that (12) becomes s ¼ 0 ¼ p tan 1 m d ; m dp þ t b t e sin 2 0 þ m d cos 2 0 ; m s ¼ s sin 2 e 0 þ t P s cos 2 0 ð14þ ð15þ ð16þ by using m d instead of m s. For example, m d = corresponds to m s = 0.3, and we obtain 0 = 39.5 instead of 0 = 36.7 as shown in Figure 8. For this angle the stress condition obtained on both branches change substantially (Table 2). [20] We show snapshots of rupture propagation in Figure 9 and phase diagrams of stress in Figure 10. Figure 9 corresponds to Figure 5b. The only difference is the direction of primary fault, 0 or 0. In this case, rupture preferred the tensional branch, as shown in Figure 9, although the fault orientation is only different by a few degrees. The reason for this change becomes clear when we focus on the initial state of the shear stress t 0, the peak strength t p, and the residual stress level t r in Figure 10 and in Table 2. In all cases, we observe that the stress increase t e required for rupture initiation is smaller on the tensional branch than on the compressional branch, although the initial shear stress is the same on both branches with respect to dynamic friction m d. Thus rupture was enhanced preferably on the tensional branch, and rupture propagation stopped on the compressional branch. Of course, we observed that in the case where the initial conditions on both branches are close to each other, rupture could propagate on both branches as shown in Figure 5c, and in the case where shear stress is not loaded enough in the branches, it is arrested around the branching point as in Figure 5a. [21] In all examples, rupture phenomena can be discussed on the basis of the initial relation between fault properties and shear stress as already modeled by Aochi et al. [2000b]. The change of shear stress is larger than that of normal stress because most of the shear stress change is due to the propagation of the rupture front. The results and discussions in this paper imply that modeling without considering the effects of normal stress may be applicable to the situation on which the Coulomb friction law is applied. The effect of normal stress is included in the frictional parameters t p and t r before starting dynamic simulation. 5. Discussion [22] The name of compressional and tensional branches is not very adequate in dynamics. From strict Coulomb friction analyses we would conclude that the tensional branch is always Figure 9. Snapshots of rupture propagation in the case m d = (corresponding to m s = 0.3). Rupture propagates along the tensional branch in contrast to the very similar case shown in Figure 5b.

8 ESE 5-8 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS Figure 10. Diagram of stress history for Figure 9 (dark lines). Symbols show the initial state. Light lines represent the previous case shown in Figure 7, curve b as a reference. more favorable for rupture because of the effect of normal stress. This implication is somewhat true as long as we examine the frictional parameters t p and t r. In fact, however, fault selection of a branched fault is entirely determined by the relation between the applied initial shear stress t 0 and the assumed frictional properties such as the peak strength t p and the residual stress level t r on each branch. Stress drop (simulation 1) or stress concentration (simulation 2) with respect to the applied shear stress is much more important than whether the fault is a compressional or tensional branch. From this viewpoint, we confirmed that our previous modeling of branched faults by Aochi et al. [2000b] without considering the role of normal stress effect might also be valid for general cases for Coulomb friction. Although we showed simulation results only for a confining pressure P = 100 MPa and a branching angle a = ±15, our results for the selection of a branched fault are very general. From Figure 4, potential branching angle strongly depends on confining pressure P and friction coefficients m s, but in every case, we observed three kinds of branching phenomena; rupture propagation on both branches, a single branch, or no branch. Our simulations imply that the value of the fault angle with respect to the potential branching angle range (circles in Figure 4) is what determines whether a fault will take a branch or not. For example, under the same confining pressure, P = 100 MPa, rupture could propagate on the compressional branch of angle a =10 when m s = 0.6, but it will never propagate onto a fault branch a =45 no matter what the value of m s. [23] As Aochi et al. [2000b] discussed, rupture propagation on both branches requires a very delicate initial condition. That is why rupture often seems to selectively propagate along only one segment in the field, although the 1891 Nobi, Japan, earthquake [Mikumo and Ando, 1976] and the 1979 Imperial Valley earthquake [Archuleta, 1984] clearly showed branching, for example. In this sense, fault selection is the same phenomena as a bending fault. Following our simulation results, whether rupture bends toward the compressional region or in the tensional direction depends on each situation surrounding the corresponding earthquake. Both ruptures which propagated along segments in the compressional and in the tensional directions were recognized [e.g., Tsukuda, 1991]. Tsukuda [1991] also suggested that rupture progress on a compressional barrier might enhance the rupture itself and lead to great damage around the fault. This implication is consistent with our simulations, since breakdown strength drop t b is always larger on the compressional branch. The 1992 Landers earthquake is an interesting example of fault selection [Aochi and Fukuyama, 2002; H. Aochi et al., Constraint of fault parameters inferred from nonplanar fault modeling, submitted to Geochemistry, Geophysics, Geosystems, 2001]. Rupture propagating northward along the Johnson Valley fault chose a branch in the tensional direction, but the fault trace as a whole bent gradually toward the compressional direction. Another example of a fault bending toward the compression field is the 1943 Tottori, Japan, earthquake where fault bendings at both ends were in the compressional direction [Kanamori, 1972]. [24] Several authors have studied the effect of normal stress on static/dynamic rupture process [e.g., Harris and Day, 1993, 1999]. In their models the main fault and the branches were not connected, while in our study the fault system is continuous. Let us consider disconnected subfaults, as in Figure 11, by adding an unbreakable barrier near the branching point. In this case, the rupture front has to jump over the barrier if rupture will propagate along the branches. This model behaves very similarly to that of Harris Figure 11. A symmetrical disconnected fault system which consists of three planar subfaults. In this case we put an unbreakable barrier on the grids whose numbers are 11, 12, 31, 32 along x 1 axis in the previous three-forked model illustrated in Figure 1.

9 AOCHI ET AL.: NORMAL STRESS EFFECT ON NONPLANAR FAULTS ESE 5-9 Figure 12. Snapshots of rupture propagation along a disconnected fault system as shown in Figure 11. We assume the frictional coefficient m s = 0.3. Since the primary fault plane is assumed to be most suitable, rupture does not jump to the other subfaults. and Day [1993, 1999]. Figure 12 shows snapshots of rupture propagation along the fault system of Figure 11, and Figure 13 shows how stress increases in the subfaults due to rupture propagation on the primary fault. Since the rupture front does not propagate directly into the branches, stress accumulation or relaxation is similar for shear and normal stresses at a distance, so that they affect the fracture criterion in similar ways. Though rupture was not yet triggered on the branches at the time step t = 150t, when the rupture arrived the end of the fault model (x 1 = 30s, x 2 = ±30s), we can see that the stress accumulation on the tensional branch due to the normal stress change is very favorable for rupture. [25] In this way, normal stress may affect rupture process as a triggering mechanics on separate faults at a distance. Thus our simulation results for a discontinuous fault system does not conflict with previous studies using FDM. The difference between continuous and discontinuous faults comes from the existence of a crack front. In the former, the rupture front arrives directly on the branched fault, so that shear stress changes are dominant. In the other case, stress concentration due to crack front disappears on a discontinuous fault, so that both shear and normal stresses may affect it in the similar way. As a result, it is very important whether fault ruptures are continuous or not, or whether rupture propagates continuously along a fault system or not, when studying dynamic rupture process along a nonplanar fault system. 6. Summary [26] We numerically simulated dynamic rupture propagation process along a three-forked symmetrically branched shear fault embedded in a simple triaxial compressional stress field and investigated the effect of normal stress by assuming a Coulomb friction law. First, we pointed out that rupture did not always prefer the tensional branch to the compressional one. Second, we found that the selection of the compressional branch depended on the direction of the primary fault plane, so that rupture could be interpreted in terms of fault properties (peak strength t p and residual stress level t r ) and applied initial shear stress on the fault t 0. The simulation results imply that the previous discussion of Aochi et al. [2000b] based only on shear stress analysis is also applicable for the current simulations including the Coulomb law. Fault selection in dynamic rupture propagation along a branched fault is determined by stress concentration near the rupture front, so that normal stress is the less important factor. Then we showed the difference between a continuous branching fault and separate fault branches. In the latter case, change of normal stress may play as important a role on rupture triggering as shear stress change for separate faults. For separate branches the mechanics is similar to static Coulomb analysis. These results demonstrate that fault continuity and discontinuity are just as important for dynamic rupture propagation as fault structure and orientation. [27] Acknowledgments. We would like to thank S. Nielsen, P. Segall, and the anonymous Associate Editor who reviewed and offered some valuable comments. Numerical simulations were made on a COMPAQ ES40 cluster (EV6 500 MHz) at Institut de Physique du Globe de Paris (IPGP). This research was supported by ACI Simulation des tremblements de Terre of the Ministère de la Recherche, France. Figure 13. Change of normal and shear stresses during dynamic rupture propagation shown in Figure 12. In this case, since abrupt change of shear stress due to the progress of rupture front does not appear, the tensional branch is closer to the fracture point. References Aochi, H., and E. Fukuyama, Three-dimensional nonplanar simulation of the 1992 Landers earthquake, J. Geophys. Res., /2000JB000061, in press, Aochi, H., E. Fukuyama, and M. Matsu ura, Spontaneous rupture propagation on a non-planar fault in 3D elastic medium, Pure Appl. Geophys., 157, , 2000a. Aochi, H., E. Fukuyama, and M. Matsu ura, Selectivity of spontaneous

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