Dynamics of Fault Interaction: Parallel Strike-Slip Faults

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. B3, PAGES , MARCH 10, 1993 Dynamics of Fault Interaction: Parallel Strike-Slip Faults RUTH A. HARRIS U.S. Geological Survey, Menlo Park, California STEVEN M. DAY Department of Geological Sciences, San Diego State University, San Diego, California We use a two-dimensional finite difference computer program to study the effect of fault steps on dynamic ruptures. Our results indicate that a strike-slip earthquake is unlikely to jump a fault step wider than 5 krn, in correlation with field observations of moderate to great-sized earthquakes. We also find that dynamically propagating ruptures can jump both compressional and dilational fault steps, although wider dilational fault steps can be jumped. Dilational steps tend to delay the rupture for a longer time than compressional steps do. This delay leads to a slower apparent rupture velocity in the vicinity of dilational steps. These "dry" cases assumed hydrostatic or greater pore-pressures but did not include the effects of changing pore pressures. In an additional study, we simulated the dynamic effects of a fault rupture on 'undrained' pore fluids to test Sibson's (1985, 1986) suggestion that "wet" dilational steps are a barrier to rupture propagation. Our numerical results validate Sibson's hypothesis by demonstrating that the effect of the rupture on the 'undrained' pore fluids is to inhibit the rupture from jumping dilational stepovers. The basis of our result differs from Sibson's hypothesis in that our model is purely elastic and does not necessitate the opening of extension fractures between the fault segments. INTRODUCTION PREVIOUS STUDIES The length of fault rupture often determines the The earliest studies of fault steps began with the magnitude of a strike-slip earthquake. It is therefore qualitative analyses and detailed field descriptions by critical to determine what controls the length of fault rupture. A geometrical feature, the fault step (Figure 1), may have controlled the rupture lengths of five recent moderate-size (J 5-6) strike-slip earthquakes in California, and a number of large (J 7) strike-slip earthquakes in Turkey. $ibson [1985, 1986] described how fault steps affected the ruptures of the 1966 (M 6.0) Parkfield, 1968 (M 6.8) Borrego Mountain, 1979 (M 6.6) Imperial Valley earthquake, and the 1979 (M 5.9) Coyote Lake earthquake [see also Reasenberg and Ellsworth, 1982]. Ryraer [1989] noted the presence and possible effect of a fault step during the 1987 (M 6.7) Superstition Hills earthquake. In Turkey, Barka and Kadinsky-Cade [1988] found evidence for fault steps and bends affecting rupture lengths of moderate to Burchfiel and Stewart [1966] and Crowell [1974]. Based on field observations of the Death Valley fault zone, Burchfiel and Stewart [1966] postulated that a zone of tension (a pullapart basin) resides between right-lateral faults which step to the right. Crowell [1974] depicted the anatomy of a pullapart basin in southern California (the Salton Trough) and also added the idea that a zone of compression (a pushup) resides between right-lateral faults which step to the left. The pioneering work contributed to the widespread identification of fault steps; however it was not until 1980 that the first quantitative studies of fault interaction were presented. Rodgers [1980] used dislocation theory to study basin development by fault steps in strike-slip faults. His analyses great earthquakes in the Anatolian fault system. The did not include the critical factor of fault interaction. interaction of fault steps and bends with smaller earthquakes Segall and Pollard [1980] did include fault interaction in has also been hypothesized. Bakun et al. [1980] suggested their two-dimensional quasi-static study of strike-slip faults. that several small (M 3-4) earthquakes on the San Andreas fault in California, including the 1973 Cienega Road They proposed that since compressional steps (left steps in right-lateral faults or right steps in left-lateral faults) earthquake were confined by fault steps. Fault steps are cause an increase in the mean stress and an increase in however observed at many different length scales in the field, from less than one centimeter to kilometers [e.g., Bartlett et al., 1981; Vedder and Wallace, 1970]. At which length scale does a fault step interact with the dynamic earthquake process and control the rupture length of an earthquake? This paper analyzes two-dimensional dynamic models of stepovers in strike-slip faults to consider this interaction. the normal stress, compressional steps should act as a barrier to rupture propagation and should act to stop earthquakes. Sibson [1985, 1986] presented the opposite viewpoint and proposed that dilational steps should act to stop earthquakes. Sibson's hypotheses were based on the interaction of pore-fluid pressure with the fault step scenario. He determined that the tensional regime created by a dilational step should act to momentarily decrease porefluid pressure. The decrease of fluid pressure would cause Copyright 1993 by the American Geophysical Union. an increase in fault strength (until the fluids re-equilibrated) and thereby act to delay or terminate the rupture. Paper number 92JB /93/92JB Mavko [1982] also modelled fault interaction using twodimensional quasi-static calculations. He used his results to 4461

2 4462 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION FAULT MAP VIEW STEPS COMPRESSIONAL STEP DILATIONAL STEP stepover width December 1939 which ruptured about 400 km at the Earth's surface on the Northern Anatolian fault [Barka and Kadinsky-Cade, 1988]) by rupturing through fault steps. We are thus motivated to analyze the problem of dynamic rupture propagation across fault steps. To examine how the dynamic stress field affects rupture on an adjacent parallel segment we consider a spontaneously propagating shear fracture in an elastic medium. Harris et al. [1991]. briefly presented results of a two-dimensional study. In this paper we have greatly expanded upon these results and discussed some ramifications. overlap METHOD To perform the numerical simulations a two-dimensional finite-difference computer program [Day and Shkoller, written comm., 1990] was used. This program is a two- SIDE VIEW dimensional version of the three-dimensional version used by Day [1982a, b]. The new program also accommodates non- I... ' coplanar fault segments. The two-dimensional case implies that one is solving the problem of plane strain. The two fault segments (shear cracks) were assigned half lengths of 14 km (Figure 2). The two fault segments were initially set at shear stress levels below failure, so the Fig 1. Right and left steps in a left-lateral vertical strike-slip fault. When two of the fault segments are slipping at the same time, a rupture on the first fault segment had to be nucleated quasiright step is a compressional step and a left step is a dilational dynamically over a distance defined as the critical radius, rc step. For right-lateral faults, right steps are dilational and left [Day, 1982b; Andrews, 1976b, 1985]. rc was taken to be steps are compressional. The stepover width is the perpendicular much shorter than the fault segment length so that it would distance between the two faults and the overlap is the along-strike distance of fault crossover. When the two fault ends do not pass not affect the dynamic results, but long enough so that it each other the overlap is negative, as shown for the compressional would permit the rupture to propagate on the first fault step. segment. Beyond r the rupture was allowed to propagate spontaneously using a slip-weakening fracture criterion [Ida, 1972; Day, 1982b; Andrews, 1976a, b, 1985] and a Coulomb friction law. The Coulomb friction law dictated that both successfully explain the creep records on the San Andreas static and dynamic friction were proportional to the normal fault near Hollister, California. Aydin and Schultz [1990] looked at the fault interaction problem in an attempt to stress. The slip-weakening fracture criterion (Figure 3) quantify the relationship between fault stepover width and enforced a linear drop in strength each time a point on a overlap for en echelon strike-slip faults around the world. fault segment first b gan to slip. (If the physics required This type of relationship had been previously determined for it, points on a fault segment were allowed to slip more than overlapping spreading centers on mid-ocean ridges (tension once.) The strength dropped linearly (in proportion to slip) cracks) by Sempere and MacDonald [1986]. Aydin and from a static yield strength to dynamic frictional strength. Schultz [1990] concluded that fault interaction is responsible The slip distance over which this strength decrease occurred for the observed en echelon fault geometry. They could not is defined as the slip-weakening critical distance, do. A do however quantify the relationship between the geometrical of 10 cm [Day, 1982b] was used. At the end of the first parameters. fault segmenthe static yield stress (cohesion) was very While of interest for understanding the static problem of fault interaction, these quasi-static analyses do not include one critical element, the time-dependence of fault rupture. 28 km. Quasi-static studies cannot include the stress waves and time-dependent stress concentrations generate during the earthquake rupture process. It is the stress field during the. -_,-. FAULT 2 dynamic rupture process which loads the next fault segment FAULT 1 and determines if the rupture can jump to the next fault T sec segment; hence a quasi-static analysis is inappropriate for assessing the likelihood on one segment of a complex fault zone developing into a larger-magnitudevent involving 28 km two or more segments. A dynamic analysis is required to Fig. 2. Map view of two faults at 3.4 s for the case of a dilational investigate the conditions under which a moderate M 5- step (left step in left-lateral shear) and the parameters listed in 6 earthquake (e.g., the July 1967 Ms Pulumur Table 1, Case A. Both faults are 28 km long. Stepover width is earthquake on the Northern Anatolian fault zone in Turkey, I km, overlap is 5 km. Open circle indicates point where rupture which had a surface rupture of about 4 km [Barka and first nucleated on fault I at 0 seconds. At 2.9 s the rupture first reached the end of fault 1. At 3.4 seconds the point marked by the Kadinsky-Cade, 1988]) could cascade into a devastating solid circle on fault 2 to rupture. After 3.4 s, the rupture M 8 earthquake (e.g., the great Erzincan earthquake of propagates bilaterally on fault 2.

3 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION 4463 Slip-weakening Fracture Criterion TABLE 1. Simulations Variables I d o slip Test Case A B C D Initial shear stress, bars Initial normal stress, bars Static coefficient of friction Dynamic coefficient of friction S P wave velocity, km/s S wave velocity, km/s Density, g/cm C rid size, km Fault i half-length, km d o- slip-weakening critical distance y = static yield stress = - $tat ( normal initial shear stress ( f = dynamic yield stress= - vn ( norma Fig. 3. The slip-weakening fracture criterion defines the strength of the fault as a function of slip on the fault. Initially the fault strength is the static yield strength. When the fault first begins to slip the strength linearly decreases to the dynamic friction strength. Once the fault has slipped a critical distance, do, the fault strength is equal to the dynamic strength. high so that the rupture could not break through into the surrounding medium. In the quasi-dynamic [e.g., Archuleta and Day, 1980] and kinematic [e.g., Hartzell and Heaton, 1983] models often presented in the literature, the rupture velocities are predetermined and fixed. The spontaneously propagating rupture velocities in our models were not predetermined, but instead were controlled by the physics of the rupture and the slip-weakening fracture criterion. Andrews [1976b, 1985], Das and Aki [1977], and Day [1982] have shown that the rupture velocity depends on the relationship between the initial stress conditions relative to failure and the stress drop. Andrews [1976b] and Das and nki [1977] defined a dimensionless parameter $ to quantify this ratio: s (i) where v is the static yield stress, 0 is the initial shear stress, f is the dynamic yield stress, and 0- f is the stress drop, and Das and Aki [1977] showed that for inplane shear cracks the rupture velocity is subshear when S exceeds 1.63 and supershear when S is less than In our modelling we chose initial conditions which would produce subshear and supershear rupture velocities to determine the effect of the rupture velocity on the results. The effects of the rupture velocity are analyzed in the discussion section of this paper. In addition to the physical parameters which defined the fault step problem (Table 1), it was also necessary to ensure that the numerical discretization of the medium would provide accurate results related to a continuum. The relevant parameters included the element size for the finite difference grid, the outer boundaries for the grid, and the Fig. 4. damping parameter for the medium. For this problem an element width of 0.25 km was selected (Figure 4). The outer boundaries of the finite-difference grid were chosen so that no waves reflected from the outer boundaries could interact with the fault segments within a reasonable time interval. The damping parameter dampened the artificial dispersion caused by the discretization of the medium. This parameter was selected to be 0.5 based on the results of previous numerical modelling. The application of our numerical modelling to dynamic rupture propagation on non-coplanar fault segments produced three scenarios which are detailed in the next section: (1) The rupture could die at the end of the first fault segment. This would lead to the shortest rupture length and therefore the smallest magnitude earthquake. (2) The rupture could trigger the second fault segment, but then run out of energy and stop propagating. This would lead to a slightly larger earthquake since the rupture would have been more energetic than in scenario (1). (3) The rupture could trigger the second fault segment, then continue to propagate. FINITE DIFFERENCE GRID fault 2 I',stepover fault fault I ; fault overlap 14 km = 56 grids grid spacing=250 m normal stress = -333 bars (compressional) shear stress = 200 bars (left lateral) width The finite difference grid used in our two-dimensional modeling. The grid is a map view of the earth's surface, so each fault is shown as a fault trace.

4 , 4464 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION This last case would produce the largest earthquake since the rupture length would be the longest. Figure 5 shows a flowchart of the modelling process which results in these three scenarios. RESULTS This section presents the results from four sets of twodimensional cases whose parameters are listed in Table 1. For each set of cases, 20 simulations were run, with the stepover widths ranging from -5 to q-5 km in 0.5 km intervals. Each of these simulations was for two shear cracks, simulating two left-lateral vertical strike-slip faults of infinite vertical extent. Test case A simulated a 100 bar stress drop (a very high stress drop) using a static coefficient of friction of 0.75 (an "average" value from Byeflee [1978]) and a dynamic coefficient of friction of 0.3. The initial shear stress was 200 bars, emulating a "weak" fault setting [Brune et al., 1969]. It took 2.9 seconds for the rupture to first reach the end of the first fault segment (a supershear rupture velocity as predicted by Das and Aki [1977] and Andrews [1985]). Rupture of the second segment then depended upon its geometrical relationship with the first fault segment. Figure 6 summarizes the location and time of the first point(s) to rupture on the second segment for the case of 5 km overlap. Note that the dilational steps triggered later than the compressional steps with equal stepover width (for stepover widths greater than 0.5 km). Furthermore the locations of the initial point of rupture on the offset fault segment differed for the dilational and compressional steps. Test case B simulated the stress drop from an earthquake with an average stress drop of 30 bars (e.g., the 1966 Parkfield earthquake [Archuleta and Day, 1980]). This simulation incorporated a static coefficient of friction of initial stress distribution PROCEDURE fault geometry] 2-D finite difference code spontaneous rupture propagation on fault 1 I rupture initiates on fault rupture propagates rupture dies rupture dies on fault 2 on fault 2 on fault 1 Fig. 5. Flowchart of the numerical modelling process used in this paper (see text for discussion) and a dynamic coefficient of It took the rupture 3.35 seconds to travel the 14 km from the middle to the end of the first fault segment (a supershearupture velocity). Figure?b shows the time-dependent rupture pattern for case B. Once again, the dilational steps generally triggered later than the compressional step with equal stepover width (for widths greater than 0.5 km) and the locations of the initial point of rupture on the offset segment differed for the dilational and compressional steps. Case B is of interest because it presents an example of "scenario 2" where the second fault segment triggered, but the rupture then died on the second fault segment. This occurred for the dilational step with a 4-kin stepover width. The rupture died on the second fault segment because the stress field was not sufficient to cause failure over a large enough patch length (on the second fault segment) to keep the rupture going. This case also demonstrates why a dynamic study is necessary. A quasi-static study would not have been able to predict that the rupture would die on the second fault segment. Test case C was similar to case A with the exception of the static coefficient of friction. The static coefficient of friction for case C was 1.1 (Table 1). This value has no widely accepted physical basis but was selected just so that the rupture velocity would be subshear. Subshear rupture velocities are often assumed by modelers of strong ground motion data, although supershear rupture velocities have been inferred from seismic observations [e.g., Kanamori, 1970a, b; Douglas, et al., 1981; Archuleta, 1982; Olson and Apsel, 1982]. For case C the rupture took 5.0 seconds to reach the end of the first fault segment. This case is interesting because although the 100 bar stress drop was the same as for case A, it appears that the difference between the initial shear stress and the static yield stress in case C was very important in determining the maximum stepover width which could be jumped (Figure 7c). Test case D had a 30 bar stress drop and a subshear rupture velocity. The rupture took seconds to reach the end of the first fault segment. The static coefficient of friction was 0.75 and the dynamic coefficient was 0.51 (Table 1). For case D the rupture could only jump dilational and fracture criterion i slip.weakening compressional A Single Fault Study steps of 0.5 km stepover width (Figure 7d). DISCUSSION To explain the location and timing of the triggering of the second fault segment for the previously described cases, we analyzed the stress perturbation due to rupture propagation on the first fault segment. It is important to realize that a single fault study only gives information about when a point on the second fault segment exceeded the static friction level and does not determine whether the stress concentration was sufficient to induce propagation on the second segment. This implies that a single fault study cannot distinguish between scenarios 2 and 3. Our models did include the rupture propagation on the second segment, therefore the single fault study (Figure 8) is presented only as an aid to understanding the rupture initiation problem. The rupture could start on the second fault segment when Coulomb friction was exceeded (when the shear stress on the fault exceeded the normal stress times the coefficient of static friction). Because both the shear stress field and

5 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION 4465 Rupture nodes 100 bar stress dro S = 0.49 static coeff. = 0.75 ic coeff. = 0.30 time (S) I ß I ' I ' I ' I ' distance (km) along strike from midpoint of fault 1 Fig. 6. Case A. Summary (map view) of the results from 20 simulations of fault steps in left-lateral shear. Table 1, Case A lists the variables used in this simulation. For each simulation only two faults exist, as depicted in Figure 2. Fault 1 is drawn with a heavy dark line. The rupture first reached the end of fault 1 at 2.9 seconds. All of the fault 2's are shown by the light parallel lines. Faults with positive stepover widths are dilational steps, negative stepover widths are compressional steps. Each solid circle indicates the point where a fault 2 is initially triggered. The times to the right of the figure are the trigger times for each fault 2. The parameter S is defined in equation (1) in the text. the normal stress field in the medium were perturbed by the propagating rupture on the first fault segment, rupture occurred on the second segment where and when the initial shear stress ( 'o) plus the change in the shear stress (/k -(t)) exceeded the static coefficient of friction (/z) times the initial normal stress (Crno) plus the change in the normal stress (/X (t)): Therefore rupture could occur when and where the timedependent stress difference, As(t) was less than zero: (2) In < o (3) Equations (1) and (2) assumed a compressional stress field (ano + Aa (t) < 0) so that the faults remained closed. Figure 8 shows As(t) due to rupture of the first fault segment using the parameters assigned to case A (Table 1). The regions with negative As(t) indicated where a second parallel segment could have begun to rupture, if a second fault segment had been present in the medium. Until the rupture reached the end of the first fault segment, the stresswaves were not sufficient to induce rupture on any parallel (non-colinear) second fault segment. As time increased, the potential rupture region developed then expanded, and more distant faults could have been triggered. This timedependent change in the shape of the potential rupture region explains the time-dependent rupture node pattern observed in Figure 6. The dynamic single-fault study also elucidated another significant detail [P. $pudich, pers. comm., November 1990]. The time-dependent stress-field, which was generated by the propagating rupture on the first fault segment, would not permit the rupture to jump to a second parallel (noncolinear) fault segment before it reaches the end of the first fault segment. Assumptions We have presented the results from two-dimensional finitedifference simulations of spontaneous rupture propagation across fault steps. The simulated faults were vertical strikeslip faults set in an otherwise linearly-elastic medium. There were many assumptions incorporated in the models, and these will now be discussed. The most significant assumption is that one can even apply two-dimensional models to 'real' faults. Obviously a two-dimensional model is not ideal; however, since this problem had not been previously solved in two dimensions, at the very least this was a very useful exercise. The twodimensional modelling will be a benchmark for comparison with future three-dimensional models [Harris and Day, work in progress] of rupture propagation across fault steps. Most importantly though, it appears that the two-dimensional models do perform a good job of predicting how some observed geometrical parameters such as stepover width and

6 4466 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION time S = O0 bar stress drop static coeff. = dynami coeff. = 0.30._, s) o_ 3.2 ;).2.'= 1.7 -o o time (s) S = 0.49.j 5 J_ bar stress drop 4! staticoeff. = J 8.3 (dies)... dynamic coeff. = '" 5.o 2 ]... -'... l 2'6!.8-1 -,2 ß o time 's)- 5.0 time(s) S = bar stress drop -- static coeff. = 1.1 '--' dynamic coeff. = S= bar stress drop -.- static coeff. = dynamic coeff. = Fig. 7. Summary (map view) of the results from 20 simulations of fault steps in left-lateral shear. This figure shows the results for Cases A-D. See Table I for a listing of the variables used in each case, and the text for a discussion. (a) Case A- same as Figure 6, this case is shown again for comparison. (b) Case B. (c) Case C. (d) Case D. fault width affect the length of earthquake rupture. Prior to this study, researchers had plotted and tabulated field observations of strike-slip fault geometry. However, even though it was suspected that fault interaction did determine some of the parameters, it was not understood at which magnitude the parameters were significant. For example, Alldin and Schultz [1990] plotted fault stepover width versus overlap for values ranging from meters to hundreds of kilometers. Bakun et al. [1980] proposed that a 200-m stepover which is observed in the trace of the San Andreas fault near Cienega Road extends down to 8 km depth and controls the ruptures of M 3-4 earthquakes. On a similar scale, Rllmer [1989] proposed that a 250 meter stepover width observed in the 1987 Superstition Hills earthquake may have affected fault rupture. Alternatively, Barka and Xadinsky-Cade [1988] and Harris et al. [1991] suggested that any two faults with stepover widths of less than i km and 0.5 km, respectively, probably merge at depth. Probably the most significant assumption in these twodimensional numerical models of fault steps is that the fault steps continue throughout the seismogenic zone. In some regions the seismicity (aftershocks) does not clearly define two distinct fault planes. On the other hand, data have indicated distinct fault planes for the San Andreas fault, as delineated by aftershocks of the 1966 Parkfield, California earthquake [yeaton et al., 1970], and distinct fault planes for the Coyote Creek fault, delineated by aftershocks of the 1968 Borrego Mountain, California, earthquake [Hamilton, 1972]. Comparison With Field Data An important point of our present study has been to determine if it were even possible for a rupture to 'jump' a fault step once the stepover width between the two faults became greater than some critical distance. This critical distance would be of utmost importance for earthquake hazard studies. An earthquake rupture which propagates

7 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION secs 4.4 secs 12.0 secs (static case) 3.4 secs 19.5 km 4.9 secs 14.0 secs (new friction value) 3.9 secs 5.4 secs 1300! 2OO STRESS DIFFERENCE (BARS) -100 Negative (dark) values rupture "-400 Fig. 8. Contoured map view of the stress difference (As) at 2.9 seconds due to rupture propagation on fault 1. Fault 1, a left-lateral fault, is the dark line. The map scale is 1:1. Negative values of As indicate regions where a second parallel fault could start but do not determine if the rupture would continue to propagate on the second fault (see text). At 2.9 seconds the rupture has just reached the end of fault 1. No negative regions exist so no parallel left-lateral strike-slip fault could trigger at this time. The parameters used in these simulations are listed down a 30 km long fault segment then stops will probably cause much less damage (by affecting a smaller area with strong ground motion) than an earthquake rupture which propagates down the same 30 km long segment then jumps across to break succeeding fault segments. If one knew the critical 'jump' distance, then one might be able to constrain the geographical region (parallel to the fault) which could be adversely affected by strong ground motion. Knuepfer [1989] notes that in his data collected from world-wide field observations of strike-slip faults, no rupture has ever jumped a compressional step wider than 5 km and no rupture has ever jumped a dilational step wider than 8 kin. Barka and Kadinsky-Cade [1988] showed evidence that the 1939 Great (M 8) Erzincan earthquake rupture on Turkey's North Anatolian fault jumped a dilational step 4-kin wide. This same rupture did not jump a 10 kinwide dilational step to the next fault segment to the north (see Figure 3 in Barka and Kadinsky-Cade [1988]). Based on their extensive studies of strike-slip faulting in Turkey, Barka and Kadinsky-Cade [1988] proposed that 5 km is probably the upper limit for the critical 'jump' distance. Wesnousky [1988] compiled data from strike-slip earthquakes in California, in Turkey (following Barka and Kadinsky-Cade [1988]), and from the great Nobi, Japan earthquake of His data (reproduced in Figure 9) showed that the Nobi earthquake jumped a 3-km wide compressional step. Therefore the field observations of Knuepfer [1989] and of Barlea and Kadinsky-Cade [1988] and Wesnousky [1988] appear to indicate that fault steps with stepover widths greater than 8 km have been jumped, and most likely 5 km is an upper limit for this critical distance. The results from our two-dimensional numerical studies show a range of stepover widths which could be jumped. Test case A which had a 100 bar stress drop (a very high average stress drop) and a supershear rupture velocity, showed that a compressional step with a stepover width of 3 km will be jumped, whereas a dilational stepover width of 3 km could be jumped. Test cases B-D showed smaller 'jumpable' stepover widths. The 'jumpable' stepover widths for the dilational steps varied from 0.5 km for the subshear rupture-velocity, 30 bar stress drop case to 5.0 km for the 100 bar supershear rupture-velocity case. Additionally, our maximum 'jumpable' compressional step was generally less than the distance for the corresponding dilational step. Knuepfer's [1989] collection of field data supports this finding. He observed that ruptures jump compressional steps with narrower stepover widths than dilational steps. Fault Lengths, Stress Drops, and Rupture Velocities A topic which also needs to be addressed is how the variables used in our simulations affected our results. These variables included the fault lengths, the stress drops, and the rupture velocities. This section will start with a discussion of fault length, which is perhaps the most easily measured geometrical parameter. In the two-dimensional

8 4468 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION lo 9- w 1 g 1 r'! compressional step dilational step 0 I I I I I I I LO 0 LO 0 LO 0 LO 0 RUPTURE LENGTH (KM) Fig. 9. Stepover width versus rupture length, modified from Wesnousky [1988]. Points indicate fault steps which were jumped during earthquakes. Data are from the 1968 Borrego Mountain, California, earthquake, 1966 Parkfield, California, earthquake, the 1891 Nobi Japan earthquake, the 1943 Northern Anatolian fault (Turkey) earthquake, and the great 1939 Erzincan earthquake, also on the Northern Anatolian fault. The earthquakes are listed in ascending order of rupture length. of simple, rectangular crack models [Day, 1982a]. For the three-dimensional problem, (and a fault of constant strength) the fault depth is often smaller than the fault length, in which case the fault depth controls the sliptime. For the two-dimensional problem, if one has a fault of constant strength along its length (no barriers), then the fault length controls the slip duration. Furthermore, it was shown by Day [1982a] that the slip-velocity and stress concentrations at the leading edge of the fault scale with fault depth in the 3-d case, while they scale with length in the 2-d case. It therefore appears that for two-dimensional models of large-magnitude earthquakes on vertical strikeslip faults, one should model the fault half-length as being roughly equivalent to the seismogenic depth. We selected a fault half-length of 14 km for this reason; 14 km is approximately the seismogenic depth for many vertical strike-slip faults. If one did want to model a fault which had a deeper seismogenic zone, then one could extend the length of the fault in the two-dimensional simulations. The result is that the 'jumpable' distance for the rupture would increase linearly with the increase in fault length [Andrews, 1976a]. It should be clear, though, that if one wanted to model a 100-kin long fault which extended from the earth's surface down to 14 km depth, then for the two-dimensional simulations the fault half-length would still be 14 kin. Next we discuss the effect of the stress drop and the rupture velocity on our results. We have presented four sets of cases, using two different stress drops, 100 bars and 30 bars. The 100 bar case corresponds to an earthquake with a very high average stress-drop and the 30 bar case corresponds to an earthquake with a moderate average stress-drop. Given supershear rupture velocities (cases A and B), the 100 bar rupture jumped a dilational step 5 km wide and the 30 bar rupture jumped 3.5 kin. Given subshear rupture velocities (cases C and D), the 100 bar rupture jumped I km and the 30 bar rupture jumped 0.5 kin. The size of the stress drop was somewhat significant in determining the stepover distance which could be jumped, but it appears that the jump distance is particularly sensitive to the model parameters which control rupture velocity. As briefly mentioned in the method section of this paper, Andrews [1976b] and Das and Aki [1977] defined a parameter, S, which determined if the rupture velocity would be super or subshear. $ relates the difference between the static yield stress (cry) and the initial shear stress (cr0) to the difference between the initial shear stress (cr0) and the dynamic yield stress (c r f), as defined in equation (1). Andrews [1976b] performed numerical experiments where he varied $ from 0.5 to 1.0 and studied the effects on inplane shear cracks. In each case the rupture velocity was supershear. Das and Aki [1977] noted that for S > 1.63, inplane ruptures travel at subshear rupture velocities and for S < 1.63 the rupture can travel at supershear rupture velocities. These results seem reasonable when one thinks fault problem, where the fault is schematically presented as a line (which represents the fault edge or crack tip), of the stress drop, cr0 -crf as the driving stress for the the fault length is the only length scale assigned to the fault. (In a three-dimensional problem the fault would also rupture. For a rupture to propagate along a fault segment, the driving stresses at the rupture front must overcome the have a depth.) This implies that for the two-dimensional difference between the initial stress, cr0, and the static yield problem, the fault length is also the shortest distance. It stress, cry. For our case A, the initial conditions (ignoring has been proposed that a point on a fault stops slipping for now changes in time) were cr0 -crf : 100 bars and (shuts off) in a time proportional to the time required for the rupture to traverse the minimum fault dimension. cry -cr0 = 50 bars. The rupture velocity was supershear. Alternatively, for case C, the conditions were cr0 -cry = 100 Numerical simulations have demonstrated that this is true bars and cry- cr0: 166 bars. The rupture in case C had the same amount of driving stress as case A, but it had to reach a higher stress level before it could continue to propagate along the fault segment. This explains why the rupture in case C travelled at subshear velocities whereas the rupture in case A travelled at supershear velocities. For both the supershear and subshear cases, the compressional steps were jumped more quickly than the corresponding dilational steps (for stepover widths greater than 0.5 km). The time delay at the dilational steps led to a decrease of the average rupture velocity when one included the time spent at the step. This could imply that field observations of rupture velocities recorded in the vicinity of fault steps are actually slower than the true rupture velocity on each fault segment. A possible example is from the Ms7.2 Dasht-e Bayaz (Iran) earthquake of Niazi [1969] noticed that the Dasht-e Bayaz earthquake had a very slow average rupture-velocity. He also noticed that the rupture was complex, possibly consisting of multiple events. Based on these two factors, Niazi [1969] postulated that the rupture decelerated at times to pause for several seconds, then resumed propagating along the fault.

9 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION 4469 Effects of Pore-fluid Pressure could be jumped by a propagating earthquake rupture. This conclusion is the same as that proposed by Sibson, although As mentioned previously in this paper, Sibson [1985, the phenomenon occurred for a different reason; in our 1986] hypothesized that dilational steps should act to stop modelling we did not include extension fractures in the propagating earthquake ruptures. He proposed that the stepover region. effect of the reduced normal and mean stress at a dilational step would be to suddenly open extension fractures. The Fault Steps and Seismic Hazard in the San Francisco Bay fluid pressure in the extension fractures would not have Region time to re-equilibrate during the time of the earthquake, and therefore the effective normal stress would suddenly increase, leading to an increase in material strength at the In the preceding paragraphs we have presented the results of a two-dimensional numerical study of rupture propagation stepover. Sibson envisioned that this scenario would at least across fault steps. We have also commented on observations temporarily delay ruptures at dilational steps. of field data from fault steps. The field data appear In this section we discuss a numerical test of Sibson's to support the applicability of our numerical results to hypothesis. Our model varied somewhat from Sibson's understanding the dynamic behavior of real faults. In this proposal in that we did not include extension fractures in the medium, but rather assumed the simpler situation of elastic material between the fault segments. As before, in section we speculate on the potential applications of our numerical results for determining earthquake magnitudes. The magnitude of a strike-slip earthquake is determined our 'dry' cases, the rupture spontaneously propagatedown part by the rupture length [e.g., Romanowicz, 1992]. One the first fault segment and the test was to determine if the rupture could jump to the second fault segment. The 'dry' cases did incorporate a static background pore-pressure in the initial normal stress, n0, but did not include timecan gather geometrical information about a step in a strikeslip fault and speculate, using our numerical models, as to whether or not a rupture could jump that step. A few cases are presented here. It is of utmost importance to realize that dependent changes in the pore-pressure (note that the values our conclusions might change significantly if the faults were we selected (Table 1) for the initial normal stress are not not separated By a fault step at depth, and/or if there were lithostatic). The 'wet' case which is discussed in this section any other faults (e.g., conjugate faults) connecting the two incorporated the dynamic effects of the propagating rupture fault segments and/or if the zone between the two faults on the 'undrained' pore-pressure where the term 'undrained' consisted of easily fractured material. Taking all of these implies that no fluid migration occurreduring the short uncertainties into consideration, we examine several striketime-scale of the earthquake. For the 'wet' undrained case slip faults in the San Francisco Bay region of California. equations (1) and (2) were modified by replacing the 'dry' Several right-lateral strike-slip fault zones extend north normal stress, an with the effective normal stress, rrn elf. and south of the San Francisco Bay area (Figure 11). rrn elf is a summed effect of the 'dry' normal stress and the dynamic pore-pressure change, These faults have been the sites of numerous moderate to large magnitudearthquakes over the last one-hundred years. Because the area is now densely populated, it has becomextremely importanto estimate the magnitudes of earthquakes which might strike the Bay area. With this goal where AP(t) is defined as in mind, a recent prediction based on creep measurements, geologic trenching data and past earthquake history [Working Group on California Earthquake Probabilities, following Rice and Cleary [1976]. B is Skempton's 1990] forecast that there is a greater than 20% chance of a large magnitudearthquake rupturing the Hayward fault coefficient, and Aakk(t)/3 is the time-dependent change in the mean stress. For plane strain, with zero strain in the y (vertical) direction, in the next 30 years. The earthquake potential of other faults in the Bay area has also been studied. The Rodgers Creek fault is the northwest extension of the Hayward fault, on the northwest side of San Pablo Bay (Figure 11). It appears AP(t) - -B[(1 q-,u)/3.](arrxx(t) q- Arrzz(t)) (6) that there is a 6-km dilational step between the Rodgers Creek fault and the Hayward fault. Additionally, the where Vu is the undrained Poisson ratio [Rice and Cleary, southern part of the Rodgers Creek fault is a 'seismic 1976]. Substituting rrn elf for rrn, the 'wet' stress-difference gap' and has been postulated as the site of a future is: M 7 earthquake [Working Group on California Earthquake Probabilities, 1990; Budding et al.; 1991]. The results of our two-dimensional numerical studies and of previous field Aswet(t) ryn ely (t)[- ]T O q- AT(t)] observations both predict that a propagating earthquake -- (rrzzo q- Arrzz(t) -- B((1 q- (7) rupture is not likely to jump a 6-km wide step. This signifies + Aezz(t))/3)[- ITO + A'(t)l that if there is a 6-km-wide dilational step between the Rodgers Creek and Hayward faults, then an earthquake where the fault-normal stress is in the z direction. Figure 10 which initiated on the Rodgers Creek fault is unlikely to shows ASwet(t) determined with a Skempton's coefficient of jump across San Pablo Bay (at the step) and to continue, 0.8 and an undrained Poisson ratio of 0.25 (close to the rupturing the Hayward fault also (a devastating scenario), values for Westerly granite listed by Rice and Clear [1976]). or vice versa. If, on the other hand, these two faults were not The consequence of including the changes in pore-pressure separate faults (e.g., if they join at a bend [Wong, 1991]), was to greatly reduce the dilational stepover distance which or if there are cross faults, then this 2-D modelling would

10 _ HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION 3.4 seconds... :':':' ::.. <a :..."! - ;: ;;'",.: :. _.:.:.-'::.:.::.:.:.:. :'.:.'.' ::'::::z: ': :W.z. :.:.:.' '.F.'>.:' :.:...::: ::::. :.:.z.: ::::::::::::::::::::::::::::....:.½ :,,, :.:...:,.'., '...'... :,,..;. ß :i.': 4 :::'.:. " /. ;:.::':' 4.4 seconds ß = ':- :'...=...:,... _...:....;','.::'.:i:..-' :. " : i!!!:!::- ' ' ' :"'.".,,, '."...' t :.-'::.:...'.:'.:': {?'..'...Y-.':."..".'i.. :!:.. ;:':.':, : ':.'::i:..."... ' :J:' i "i:' i:<...., , ; seconds 12.0 seconds ' ":"' " ':"''"'.:'.: : '.".'.-'i... '"'"'"""' '..-:": ;ii:' iiiii!:... ": : i?:i::i;i...';.l:.!:.':... :'...::?. '":':'":'"'"c<.' ¾. ":':" i: :i. ß... ; ::.-:':.::'.'..'::, :-_:.-:'.'s:::: -[.:....:.,. " : Fig. 10. Contoured map view of the stress difference (As) at 3.4, 4.4, 5.4, and 12.0 seconds for the case which (a) does not include the effects of changes in pore pressure (same as Figure 8), and (b) does include the effects of changes of pore pressure. Light regions indicate where a second parallel fault could trigger. Note that when the effects of changing pore pressure are included the rupture has difficulty jumping dilational steps. _ 38 ø - PACIFIC OCEAN _ -- not apply and the problem must be reformulated. The role of complex connecting structures will be addressed in the future with further numerical modelling. Another Bay area example is the Concord fault. Its northern counterpart is the Green Valley fault, and a map view shows a 2- to 3-km wide dilational step at Suisun Bay, between the two faults. Oppenheimer and MacGregor- Scott [1991] stated that there is a 10 km seismic gap on the Concord fault north of the city of Concord and that this gap could fail in a M 5.5 earthquake. If this hypothesized M 5.5 earthquake jumped the dilational step and ruptured the Green Valley fault then this earthquake could become a much larger event by greatly increasing its rupture length. It is however more difficult to speculate on whether or not an earthquake will jump a 2-3 km wide dilational step. There is numerical and observational evidence that dilational steps of this width have both stopped earthquakes and allowed them to continue propagating. In this light, we decline to speculate on whether or not the Concord-Green Valley fault step could be jumped. SUMMARY AND CONCLUSIONS I ' ' ' ' I ' ' ' ' 123 ø 122 ø Fig. 11. Loc tions of f u]t steps in the S n Francisco B y region of C ]iforni. The [ext discusses the right step between the right- ] ter ] strike-slip Hayward nd Rodgers Creek f u]ts, nd the righi s[ep be[ween [he rigbi-lo[ertl s[rik s]ip Concord nd Green V lley f ults. To summarize, we have used two-dimensional numerical modelling to simulate dynamic rupture propagation across fault steps. Previous non-dynamic studies had presented opposing viewpoints, one stating that a compressional fault step would stop an earthquake, the other that a dilational step should stop it. Our results have shown that earthquake ruptures can spontaneously propagate across both dilational and compressional fault steps, concurring

11 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION 4471 with field observations of strike-slip faults. The maximum stepover width which can be jumped by a propagating rupture appears to depend on whether or not the rupture velocity on the first fault segment is supershear. More precisely, the same model parameters which control rupture velocity also influence the maximum stepover width. In particular, lower values of the dimensionless ratio q permit larger stepovers to rupture, as do higher stress drops. The numerically modelled dilational fault steps delayed the rupture velocity for a longer time than the compressional steps did, leading to a lower apparent rupture velocity. This phenomenon, of dilational steps slowing the rupture velocity, may be detectable in future strong ground motion recordings near fault steps. We also tested the effects of the dynamically propagating rupture on the undrained pore fluids in the vicinity of a fault step. Our results appear to numerically validate Sibson's observation [1985, 1986] that dilational steps delay (or stop) earthquakes, although our physical model is different from and simpler than what Sibson proposed. Our two-dimensional dynamic results have provided a tantalizing explanation for many previously unexplained phenomenon. We have explored the two-dimensional problem of fault steps and the dynamic rupture process. In our next study we will tackle the three-dimensional problem of how fault steps interact with spontaneously propagating earthquake ruptures. Acknowledgments. This study could not have been performed without the generous assistance of Boris Shkoller at S-Cubed. Dave Schwartz (USGS) pointed out the potential significance of the step between the Hayward and Rodgers Creek faults. This manuscript was internally reviewed by Joe Andrews and Bill Bakun at the USGS. 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12 4472 HARRIS AND DAY: DYNAMICS OF FAULT INTERACTION constituents, Rev. Geophys., 1J, , Rodgers, D. A., Analysis of pull-apart basin development produced by en echelon strike-slip faults, in Sedimentation in Oblique-Slip Mobile Zones, edited by P. F. Ballance, and H. G. Reading, Spec. Publ. int. Assoc. Sedimentok, 4{, 27-41, Romanowicz, B., Strike-slip earthquakes on quasi-vertical transcurrent faults: Inferences for general scaling relations, Geophys. Res. Left., 19, , Rymer, M. J., SurfaCe rupture in a fault stepover on the Superstition Hills fault, California, U.S. Geol. Surv. Open File Rep., , , Segall. P., and D. D. Pollard, Mechanics of discontinuous faults, J. Geophys. Res., 85, , Sempere, J.- C., and K. C. MacDonald, Overlapping spreading centers: Implications from crack growth simulation by the displacement discontinuity method, Tectonics, 5, , Sibson, R. H., Stopping of earthquake ruptures at dilational fault jogs, Nature, 316, , Sibson, R. H., Rupture interactions with fault jogs, in Earthquake Source Mechanics, Geophys. Monogr. Ser., vol. 37, edited by S. Das, J. Boatwright, and C.H. Scholz, AGU, pp , Washington, D.C., Vedder, J. G., and R. E. Wallace, Map showing recently active breaks along the San Andreas and related faults between Cholame Valley and Tejon Pass, California, U.S. Geol. Surv. Misc. Field Invest. Map, 1-57, scale 1:24000, Wesnousky, S. G., Seismological and structural evolution of strikeslip faults, Nature, 335, , Wong, I. G., Contemporary seismicity, active faulting and seismic hazards of the Coast ranges between San Francisco Bay and Healdsburg, California, J. Geophys. Res., 96, 19,891-19,904, Working Group on California Earthquake Probabilities, Probabilities of large earthquakes in the San Francisco Bay region, California, U.S. Geol. Surv. Circ. 1053, 51 pp., S. M. Day, Department of Geological Sciences, San Diego State University, San Diego, CA R. A. Harris, U.S. Geological Survey, 345 Middlefield Road, MS/977, Menlo Park, CA (Received December 2, 1991; revised September 11, 1992; accepted September 16, 1992.)