Properties of large ruptures and the dynamical influence of fluids on earthquakes and faulting

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B9, 2182, doi: /2000jb000032, 2002 Properties of large ruptures and the dynamical influence of fluids on earthquakes and faulting S. A. Miller 1 Geology Institute, ETH-Zurich, Zurich, Switzerland Received 30 October 2000; revised 2 March 2002; accepted 7 March 2002; published 11 September [1] A model that couples large-scale plate motion loading with the dominant mechanical effects of fluids within faults shows that the system always evolves to a high degree of disorder in the fault zone stress state. This disorder comes about because of the additional degree of freedom in stress space allowed by including coseismic changes in the fault zone pore pressure. For a given set of initial conditions the long-term fault-averaged stress state equilibrates, resulting in effective properties of the system while maintaining continuous complexity. Stress state disorder organizes in physical space into spatially correlated regions of incipient failure. If the spatial extent of incipient failure is limited, small events occur and the system continues forward. Aperiodically, incipient failure covers a large portion of the fault, and failure of a single cell cascades into very large model earthquakes. A switch from static to dynamic friction at the onset of slip results in a slip-weakening model from stress redistribution during a (quasi-static) propagating elastic dislocation. The complexity inherent in the model is quantified with simple relationships for the average stress drop, average slip, and seismic moment of large events. The characteristic length relating average slip to stress drop is shown to be asymptotic and a function of the aspect ratio of the rupture and the degree of fault overpressure. The model shows that average slip increases with rupture length but at a reduced rate with ongoing slip, showing a smooth transition from L model to W model mechanics for very large events. Model catalogs are compared with the historical catalog of strike-slip events [Wells and Coppersmith, 1994], and the good agreement between the model and observations of seismic moment and fault area shows that the model successfully predicts the average slip over rupture areas up to about 2500 km 2. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 7230 Seismology: Seismicity and seismotectonics; 8123 Tectonophysics: Dynamics, seismotectonics; 7260 Seismology: Theory and modeling; 1869 Hydrology: Stochastic processes; KEYWORDS: earthquakes, pore pressure, fault strength, slip complexity Citation: Miller, S. A., Properties of large ruptures and the dynamical influence of fluids on earthquakes and faulting, J. Geophys. Res., 107(B9), 2182, doi: /2000jb000032, Introduction 1 Now at Institute of Geophysics, ETH-Zurich, Zurich, Switzerland. Copyright 2002 by the American Geophysical Union /02/2000JB000032$09.00 [2] It has long been known that high pore pressures have a strong mechanical influence on rock and soil by simply counteracting the normal force on a frictional interface, and thus reducing its frictional resistance. The limiting cases for pore pressures in the crust are hydrostatic and lithostatic, resulting in either strong or weak frictional properties, respectively. In stable continental crust, in situ bulk permeability on the order of k m 2 to m 2 enables diffusion processes to erase any overpressure that may develop, thus resulting in hydrostatic pore pressures over long timescales [Manning and Ingebritsen, 1999; Townend and Zoback, 2000]. Hydrostatic pore pressures imply a strong crust with frictional properties consistent with laboratory measurements. However, this general state of hydrostatic pore pressures within stable crust cannot be extended to mature fault zones because permeability for highly worn fault gouge and clays that make up the fault zone core is lower by orders of magnitude, with k m 2 to m 2. Such low permeability limits diffusive processes, so given a fluid source, overpressures may develop and may be maintained over timescales of the seismic cycle. The dominant mechanism for fluid flow within and normal to the fault zone is most likely controlled by large-scale coseismic permeability changes during dilatant fracture. This is consistent with evidence from studies of fault zone architecture that demonstrate faults act as seals during seismic quiescence and as conduits during earthquakes [Chester et al., 1993; Caine et al., 1996]. Therefore overpressured faults embedded within a hydrostatically pressured crust results in weak faults, strong crust, and two hydraulically isolated systems that become connected at the time of the earthquake. [3] The importance of fluids on earthquakes and the faulting process has long been recognized [Nur and Booker, ESE 3-1

2 ESE 3-2 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING 1972; Sibson, 1973; Rudnicki and Chen, 1988; Byerlee, 1990; Rice, 1992; Blanpied et al., 1992; Sleep and Blanpied, 1992; Lockner and Byerlee, 1995; Segall and Rice, 1995; Miller et al., 1996; Yamashita, 1998], but model developments have been limited by the computationally restrictive range of timescales between wave propagation and diffusion. The simple conceptual model of fault valve behavior [Sibson, 1992] established the link between pore pressure changes and tectonic loading and is supported by a wide spectrum of observations. Fault valve behavior describes the earthquake cycle as a system where pore pressures increase within the fault zone during tectonic loading and decrease at the time of the earthquake from increased permeability associated with dilatant fracture. The numerical equivalent of this scenario [Miller et al., 1996, 1999] showed that the first-order effect of including pore pressure changes was to create disorder in the fault zone stress state. The complex stress state resulted in asperity formation and destruction and various degrees of fault zone overpressure. The model combined elastic dislocation theory with a simple model for fault zone hydraulics [Miller and Nur, 2000], thus including the dominant effects of the earthquake process. In those models, only static friction was considered and limited the size of the model earthquakes that could be generated. In this model a slipweakening model for friction is introduced that allows slip to continue at a lower stress level, resulting in continued slip and thus large model earthquakes. [4] The purpose of this paper is to explore the long-term evolution and rupture properties of spontaneously generated large model earthquakes. A series of simulations with different initial conditions demonstrate that the overall behavior of the model is independent of input conditions and that the complexity inherent in the model is quantified in simple terms. A further purpose of this paper is to demonstrate model support for the relationship between the average slip of a rupture and the rupture length hypothesized to explain earthquake scaling relationships [Miller, 2002]. 2. The Model [5] The model is described in detail by Miller et al. [1999], but a brief summary is presented here for completeness. The model is the numerical equivalent of the fault valve model [Sibson, 1992] and consists of a vertical strike-slip fault embedded in a three-dimensional elastic half-space. The fault is loaded by a displacement along the downward continuation of the fault plane, moving at plate velocity V pl. Periodic boundary conditions are placed on the vertical boundaries of the fault plane [see Rice, 1993, appendix]. The fault plane is discretized into a matrix of cells that act as both rectangular dislocations and sealed compartments. The two main components are the shear stress (t 12 component of the stress tensor) and the effective normal stress (s e = s m P p, where s m is the lithostatic stress and the assumed normal stress on the fault and P p is the pore pressure). [6] Shear stress is calculated with elastic dislocation theory [Chinnery, 1961; Stuart et al., 1985; Ben-Zion and Rice, 1995; Miller et al., 1996] and increases from plate motion via t i ¼ G 2p X N j k ij d j V pl t ; ð1þ where t i is the shear stress on cell i from a slip d on cell j, V pl is the plate velocity, G is the shear modulus, and k ij is the stiffness matrix [Rice, 1993]. Simultaneously, a fluid source (i.e., fault compaction, dehydration, etc.) within a zero permeability fault zone increases pore pressures at rates described j noflow ¼ _ f _ plastic i ; ð2þ f i b i where i is the cell matrix index, ( _ _ f plastic ) is a source term, and the compressibility has been lumped into a single parameter b = b f + b f [Segall and Rice, 1995], where b f and b f are the pore and fluid compressibility, respectively. [7] The simultaneous increase in t 12 and decrease in s e bring a cell to the Coulomb failure condition (t fs = m s s e ), where m s is the static friction coefficient. When a cell reaches this condition, the shear stress is reduced by an assumed percentage of the stress on that cell (discussed in section 2.1), and the cell slips by an amount d j ¼ t a k ii ; ð3þ where k ii is the cell self-stiffness and t a is an imposed (assumed) stress drop when a cell slips. A value of 25% of the shear stress on that cell when it slipped was usually chosen, although other values were also investigated. Elastic stress transfer to all other cells in the matrix is then t i ¼ G 2p X N i k ij d j : [8] If stress transfer is sufficient to initiate failure on other cells, then they also slip and the cycle is repeated. In a previous version of the model [Miller et al., 1999] the assumed stress reduction when a cell slipped was large (e.g., 80%), and a cell needed to be reloaded to t fs in order to slip again. This limited the size of the earthquake that could be generated. In this model, once a cell has slipped (during the same time step), it must only be reloaded to the dynamic failure condition (t fd = m d s e ) to continue to slip, where m d is the dynamic friction coefficient. Each time it reaches t fd, equations (3) and (4) are applied, and the model is cycled until all cells are below either t fd or t fs. [9] Pore pressures are redistributed coseismically to neighbor cells nearest to that which slipped as described by Miller et al. [1999] and are redistributed during propagation of the rupture. Considering the timescales for crack propagation and diffusion, a more plausible scenario is a separation of rupture and fluid redistribution. This approach is incorporated into the general model of Fitzenz and Miller [2001]. Fluid flow normal to the fault zone is not currently considered but probably contributes significantly to coseismic pore pressure changes because of a high permeability damage zone adjacent to the low-permeability fault zone core [Caine et al., 1996; Faulkner and Rutter, 2001; Tanaka et al., 2001] Friction [10] The friction model is not conventional and requires some discussion. The model used is a simple switch from a static friction coefficient (m s ) to a dynamic friction coef- ð4þ

3 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING ESE 3-3 Figure 1. Example of the stress state and friction behavior for a large model earthquake to show how the model behaves. (a) Just prior to a large model earthquake, the stress state is highly disordered in stress space but spatially correlated along the fault zone into a large region of incipient failure. Stress transfer from slip on a single cell then cascades through the stress state by a quasi-static propagating dislocation (b) before resetting at a new stress state (c) following the event. (d) The stress path of one cell involved in the rupture shows the slip-weakening behavior of the model. See text for more discussion. ficient (m d ) at the onset of slip. Although this is not an explicit slip-weakening constitutive law, it mimics slip weakening simply from the properties of a propagating (quasi-static) elastic dislocation (Figure 1). That is, when a cell reaches the static failure condition (t fs = m s s e ), the shear stress is reduced an assumed percentage of the shear stress on that cell at failure (say 25%). The self-slip from this stress change (equation (3)) is then used to calculate stress transfer (equation (4)) to all other cells. The failure condition for this slipped cell then becomes t fd = m d s e for the remainder of the time step. If stress transfer (equation (4)) is sufficient to load other (unslipped) cells to t fs or (slipped) cells to t fd, then they also slip, equations (3) and (4) are again applied, and they adopt the failure condition t fd. This cycle is repeated until all unslipped cells are below t fs, and all slipped cells are below t fd. The failure condition is reset to t fs for all cells for the subsequent time step. [11] Figure 1 demonstrates the stress state and friction behavior for a large model earthquake to show how the model behaves and to introduce the physical meaning of the terms used in the text. The stress state just prior to a large event (Figure 1a) is highly disordered in stress space through the dynamical interaction between shear stress and effective normal stress [Miller et al., 1999]. Although disordered in stress space, the stress state is ordered along the fault plane into regions of incipient failure. (Stress state evolution is described in section 3.1.) If the region of incipient failure is large, slip on a single cell cascades through the evolved stress state and generates large model earthquakes. When exactly one point in the cloud of Figure 1a (the hypocenter) reaches t fs = m s s e, shear stress is reduced and loads all other cells. The hypocentral cell is precisely constrained by t fs, but the loaded cells are not, resulting in a slight overshoot of t fs (Figure 1b). In the same way, when a slipped cell with failure condition t fd is reloaded by the propagating rupture, it too can overshoot t fd by a small amount. The rupture terminates when all cells are below either t fs (unslipped) or t fd (slipped), with the new stress state on the fault shown in Figure 1c. The next time step is determined by the time required for precisely one point in the cloud of Figure 1c to satisfy t fs. The change in stress (t s ) is small, so the model is driven forward at approximately the same absolute stress. Similar complex stress states exist in all of the simulations, with fault pore pressure defining the extent of the stress space. Additionally, Figure 1c cycles back to Figure 1a prior to the next large event, so model earthquakes from a particular realization rupture similar stress states. [12] Following the path of one point in Figure 1a involved in a large model earthquake demonstrates the slip-weakening

4 ESE 3-4 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING Table 1. Input Parameters and Evolved Properties of the Model Realizations a 1 P Model V pl, mm/yr W,km m s m d _f, yr t, MPa/yr t, MPa s e,mpa l f, % C 1 C 2 C 3 A e B e C e D e E e F e H e a Stress drop at slip of a cell was fixed at 25% of the shear stress (t) on that cell (e.g., t a = 0.25t) in all cases except model D, where t a =0.8t. The fault plane measured 340 km along strike in all cases except model F (200 km), and the initial dimensionless overpressure l o = 0.7 and G = 30 GPa in all cases. Parameters f o and b were 0.05 and MPa 1, respectively. The source term _f was uniformly distributed between the values shown in the table; read 0 1.3e 5 as Evolved properties (middle columns) were extracted when the system had reached a dynamic equilibrium (i.e., Figure 2). The fitting parameters C 1, C 2, and C 3 in the last columns are discussed in the text. behavior of the model (Figure 1d). The shear stress prior to arrival of the oncoming rupture is t o. When the rupture arrives, shear stress increases from t o to C 1 m s s e, where C 1 is the degree of overshoot of the static failure condition. Since the initial failure condition is now satisfied, shear stress is reduced an assumed percentage of the stress on that cell (25% in this case, see Table 1), and the cell slips (equation (3)) an amount d (dashed line) and loads all other cells (equation (4)). This can trigger slip on a nearby cell, whereby stress transfer back to the slipped cell reloads it above t fd. Since the dynamic failure condition is satisfied, stress drops t a and the process continues. As the rupture front passes and propagates along strike, this cell may be reloaded several times from the propagating dislocation to a value above the dynamic failure condition (m d s e ), incrementally slipping each time the condition is met. The cell stops slipping when stress transfer is no longer sufficient to reload to t fd. The cell then has a total slip u for the event. Figure 1d shows that the sliding stress is approximately m d s e and that the magnitude of the stress drop is t d model t d ideal. Therefore t d can be quantified in conventional terms as t d =(C 1 m s m d )s e, where C 1 is a measure of the overshoot of t fs. The simple switch used to approximate the change from static friction to dynamic friction shows model behavior is slip weakening with the added benefit of knowing both the stress change and the absolute stress on the fault. [13] The model produces a direct effect of increased friction due to dynamic loading, followed by slip-weakening behavior with ongoing slip. The frictional properties are thus consistent with laboratory observations simply through the calculation of rupture as a propagating elastic dislocation. The direct effect has a physical basis in that a finite breakdown time is required at the rupture front, allowing stress to build beyond it static strength. Although the simple switch friction model is crude, the actual stress-slip behavior mimics ideal slip weakening quite well. All cells in a rupture undergo similar paths, resulting in a catalogs of model earthquakes quantified by t d and u This is discussed in section Assumptions and Model Input [14] Brittle crustal faulting is very well approximated with elastic dislocation theory and is used in numerous models [Massonnet et al., 1993; Stein, 1999], so discussion in support of this aspect of the model will not be expanded here. The dominant assumption about fault zone hydraulics is that it is controlled by two distinct and extreme states: impermeable during quiescence and highly permeable to neighbor cells at slip. Although a simplification, it captures the dominant aspects of the fault valve model, and inferences to it from numerous investigations of fault zones argue that it is essentially valid [Chester et al., 1993; Caine et al., 1996; Cox, 1995]. [15] During quiescence, when permeability is everywhere zero, fluid flow is restricted both within the fault zone and normal to it. When a cell slips, new dilatant cracks form and hydraulically link the local neighborhood, allowing pressure equilibration among neighbor cells. It is assumed that this occurs instantaneously (k 1). Actually, this occurs over a diffusion timescale and length scale, so it is useful to quantify this approximation. The timescale for diffusion (t d ) over a diffusion length (l) is t d ¼ l2 k ¼ fb f hl 2 ; ð5þ k where h is the fluid viscosity ( Pa s), k is the intrinsic permeability of the matrix, b f is the fluid compressibility, and k is the hydraulic diffusivity (k = k/ hfb f ). Bounds on the degree of coseismic permeability changes are not well constrained, but estimates can be made. For fault zone permeability on the order of k m 2 to k m 2 and using the model input parameters for f and b f, the timescale for diffusion is between 60 and 400 years over a diffusion length on the order of the cell size (1 km). This is much greater than the recurrence interval for slip on a cell, so the approximation of limited (interseismic) diffusion processes within the fault zone is justified. Conversely, if it is assumed that coseismic permeability changes that accompany (dilatant) slip are on the order k m 2, then diffusion timescales range from a few days to a few months. This is approximately the range of time steps found in the model to initiate the next event. Therefore, on first order, the toggle-switch permeability [Miller and Nur, 2000] assumption is justified for small events. It should be noted that the fault zone is sealed from the surrounding crust which are purported to be under hydrostatic pore pressure conditions [Townend and Zoback, 2000; Manning and Ingebritsen, 1999], with (by implication) permeability of m 2. Hydrostatic pore pressures in stable crust would be expected because no mechanism exists to substantially reduce porosity or permeability, therefore maintaining the diffusion timescale.

5 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING ESE 3-5 Figure 2. (a) Time history of the fault-averaged stress state (model A) for a typical 10,000 year simulation. From the low-stress initial conditions the shear stress t increases owing to plate motion and decreases during earthquakes. A fluid source decreases the effective stress (s e ), while dilatant slip increases it. The ratio of pore pressure to normal stress (l) mirrors the effective stress and reaches an equilibrium after a few thousand years. Although the system has effectively equilibrated, a close-up (identified by the box in Figure 2a) shows a high degree of complexity from the generation of earthquakes of all sizes. Long-term equilibrium was typically reached around 5000 years, and the model catalogs were compiled for events in a year time window. The long-term stress states (t and s e ) listed in Table 1 were taken when the model had reached the long-term equilibrium. [16] Well-constrained model input includes the plate velocity V pl, pore and fluid compressibility (b = b f + b f ) [David et al., 1994; Segall and Rice, 1995], and the shear modulus of the elastic solid (G). Less known, but still reasonably well constrained parameters include an initial assumed porosity f and a dilatancy factor b m that relates the amount of energy available to create new crack porosity [Sleep, 1995]. The important unconstrained parameter in the model is the fluid source term _ f _ plastic. This term includes a direct fluid source _ [Wong et al., 1997] that represents, for example, a dehydration source and a source due to plastic pore closure f _ plastic [Walder and Nur, 1984; Segall and Rice, 1995]. The first source _ could be significant within fault zones from a fluid source at depth [Rice, 1992] or dehydration of hydrous minerals from frictional heating. This term is certainly important in subduction zones where saturated sediments are subducted with the slab and hydrous minerals dehydrate during subduction. The second source term ( f _ plastic ) can represent, for example, plastic pore closure due to fault compaction [Sleep and Blanpied, 1992] and pressure solution. For simplicity in evaluating the model the source term is combined into a single parameter ( f _ ¼ _ f _ plastic ) that results in a range of pore pressure increase rates (Table 1). The assumed pore pressure increase rates are in close accordance to the values determined in a full study of creep compaction by Sleep and Blanpied [1992] (D. D. Fitzenz, personal communication, 2001). Note that the choice of pore pressure increase rates does not affect the properties of the ruptures because the ruptures spontaneously arise within the surrounding stress state and propagate as elastic dislocations. It therefore does not matter, from a logic viewpoint, how the pore pressures may have developed, what matters is that the rupture must propagate through the stress state within which it finds itself. 3. Model Results [17] A series of different model realizations are presented with input parameters listed in Table 1. All model parameters were varied to demonstrate that the model behavior is general and to show that the model can be quantified by simple expressions Evolution to Effective Properties [18] The dynamical interaction between t and s e is complex [see Miller et al., 1999], but for a given set of initial conditions the model evolves to a long-term dynamic equilibrium in the average stress state of the system (Figure 2). Shear stress development along the fault plane increases from approximately zero (initial conditions) to a long-term average maintained by the competing effects of shear stress

6 ESE 3-6 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING Figure 3. Stress state through which model ruptures propagate for each model realization. This stress state is achieved by the dynamical interaction between shear stress and pore pressure (e.g., effective normal stress) changes [see Miller et al., 1999]). The rate that a cell approaches the failure condition in the t direction depends on the plate velocity V pl ), the depth z, and the shear modulus G. The rate that a cell approaches the failure condition in the s e direction depends on the source rate ( _ f _ plastic ), the porosity f, and the compressibility b. The line to the x and y axes marks the average shear and effective stress of the entire fault. The stress state of all cells in the model at any instant once the model has achieved dynamic equilibrium is shown. Although the dots are constantly moving around, the overall stress state is stable. The stress state for models B and E were taken just following a large rupture, so a large section of the fault is below the dynamic failure condition. increase with constant plate motion and shear stress reduction during model earthquakes. The fault-averaged stress state, described by t and s e, has an overpressure l defined as the dimensionless pore pressure (l = P p /s m ;0.37<l <1), where the lower and upper limits are hydrostatic and lithostatic, respectively. The overall stress state equilibrates to an average, but closer inspection (Figure 2b) shows a high degree of complexity that reflects the disordered stress state along the fault and the generation of events of all sizes and a b value of 1 [Fitzenz and Miller, 2001]. Long-term equilibrium was typically reached around 5000 years, and the catalogs of events discussed below were compiled for events in a year time window. The different equilibrium states (Table 1) depend on pore pressure buildup rate relative to the plate velocity. Increasing this ratio results in an overpressured (weak) fault, while reducing this ratio leads to an underpressured (strong) fault. It also depends on the width of the seismogenic layer, the assumed friction coefficients, and the assumed stress reduction at when a failure condition is satisfied. All of these parameters were varied to show that the expressions derived below are general for this model. [19] Figure 3 shows typical stress states of the different model realizations through which model ruptures propagate (see also Figure 1a). When a particular model realization has reached long-term equilibrium (i.e., Figure 2), extracting the stress state at any particular time looks like Figure 3. Although continuously changing on a local scale, the overall stress state remains relatively stable. This produces a wide-range of rupture sizes for the same fault model because the clouds in Figure 3 correlate into patches of incipient failure of different sizes along the fault. In all cases, the stress space is filled by the dynamical interaction of pore pressure changes and dilatant frictional sliding. Complexity in the fault zone stress state is thus very easy to generate because the extra degree of freedom in the s e direction provides continuous mixing of the stress state. That is, stress changes in both the t and s e directions, whereas friction-only models are limited to the t direction. As a consequence, friction-only models require either highly rate-dependent friction [Cochard and Madariaga, 1996; Shaw and Rice, 2000] or statistically complex a priori assumptions of along-strike frictional properties [Ben-Zion and Rice, 1995] to generate complexity. [20] All events in the model are initiated by slip on a single cell, so hypocenters are true model hypocenters. If the hypocenter occurs in a region of isolated failure, then this patch fails, stress and pore pressures are redistributed, and the model marches forward. Aperiodically, the region of incipient failure becomes correlated through the sum of all previous events, resulting in a large region of the fault on the verge of failure. All that is required then is a trigger that sets the rupture on its way. Figure 4 shows the model earthquake that corresponds to the discussion of Figure 1. Figure 4a (top

7 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING ESE 3-7 Figure 4. Rupture corresponding to the stress state in Figure 1a. The top panel in Figure 4a shows how the stress state in Figure 1a is spatially distributed along the fault plane and shows that a large region is near the static failure condition. The top panel in Figure 4b shows the slip deficit distribution (relative to the plate position). The sequences that follow show (a) contours of initial stress drop and (b) slip behind the rupture front. Slip on a single cell near middepth propagates through the evolved stress state, initially expanding as a circular rupture before saturating at the fault width and propagating along strike. Slip accumulates behind the rupture front because of stress transfer to regions sliding at the dynamic shear stress and shows how slip increases with rupture length. Scale bars are slip deficit (28 32 m), slip (0 4 m), initial stress drop (0 20 MPa), and proximity to slip t/s e > panel) shows how the stress state in Figure 1a is spatially distributed and shows the proximity to slip just prior to a M w 7.5 (M o = N m) model event. The sequence that follows shows the rupture mapped as contours of stress drop at the rupture front (e.g., the initial stress drop of a cell and the overshoot of m s s e in Figure 1b). Figure 4a (bottom panel) shows how the stress state in Figure 1c is spatially distributed along the fault after the event. This particular event begins near middepth of the fault and expands as an approximate circular rupture until it saturates at the fault width. It subsequently ruptures along strike with accumulating slip behind the rupture front (Figure 4b). Increasing slip with rupture propagation results from stress transfer at the rupture front to the failed cells sliding at t fd. The rupture terminates simply because the stress state to which it is trying to propagate is not sufficient to propagate the rupture. Stress triggering arguments are seen visually by the subsequent incipient failure along the fault plane immediately following termination of the rupture (Figure 4a, bottom panel). Shear stress has been enhanced, thus advancing the time to rupture along this part of the fault. Whether the triggered event is large or not depends on the stress state. In this case, the stress state is advanced to produce a moderate sized event but will not propagate a significant distance because patch of incipient failure is isolated Synthetic Catalogs [21] The models generate hundreds of thousands of events like in Figure 4, and synthetic catalogs were compiled for each model realization. Rupture properties extracted from the catalog include the seismic moment (M o ), the average dynamic stress drop (t d ), the average slip (u), and the area (A) that slipped. The rupture area is known for small events, but the geometry can be complex, so it is difficult to distinguish length (L) from width (W ). Thus, for events smaller than the width of the seismogenic layer I assume L W. Large events are approximately rectangular, so L, W, and Z (the mean depth of rupture) are well constrained. Figure 5 shows the slip distributions for the largest event in the catalog for each model realization. All events are initiated by the failure of a single hypocentral cell (marked with a cross), and the ensuing failure from a propagating elastic dislocation results in complex slip distributions in all cases Stress Drop and Slip [22] Despite the complexity inherent in the model, it is quite stable, and the model catalogs can be easily quantified by simple relationships. The derivation below follows the logic used by Miller [2002] to argue that fault overpressure is the scaling parameter for large earthquakes. An important link in that argument was the form of the characteristic length that relates the average slip of an earthquake to the stress drop. Below I show that the proposed form for the characteristic length is supported by the properties of large ruptures in this model. [23] Assuming that the normal stress on a fault is the lithostatic stress, then s e = s m P p = rgz (1 l), where r r is the rock density, g is gravity acceleration, Z is the mean depth of the rupture, and l is the dimensionless pore pressure. A model earthquake initiates when one point in the cloud of

8 ESE 3-8 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING Figure 5. Final slip distributions of large model earthquakes from each of the models considered. Model earthquake arise spontaneously by the cascading failure (see Figure 1) initiated by the hypocentral cell ( X ). The slip wraps around to the other side of the fault from the periodic boundary conditions. Notice the change in scale for the color bar. The models generate many thousands of such events, and synthetic catalogs were compiled for each model. Rupture properties extracted include the seismic moment (M o ), the average dynamic stress drop (t d ), the average slip (u), and the area (A) that slipped. Figure 1a or Figure 2 reaches the failure condition t fs ¼ m s s e ¼ m s r r gzð1 lþ: ð6þ Once sliding begins, it continues at the dynamic failure condition t fd, t fd ¼ m d s e ¼ m d r r gzð1 lþ ð7þ with the difference defined as the dynamic stress drop t d t d ¼ t fs t fd ¼ ðc 1 m s m d Þr r gzð1 lþ; ð8þ where C 1 is a measure of the overshoot of the static strength due to the breakdown process at the rupture front (see Figure 1d). Equation (8) shows that t d is a function of Z for faults of the same overpressure (l). For large earthquakes (e.g., Z! 1 2 W), t d is constant as L grows large. Figure 6 compares the synthetic catalog with equation (8) for model realizations as a function of depth and for all models of surface breaking ruptures in Figure 7 as a function of L. Equation (8) shows that t d is constant as earthquakes grow large for a given fault overpressure l. This is understood by considering the average stress state of large events (Figure 3). It was shown that the dynamic equilibrium is steady through time, so events must propagate through a fault with average effective properties. When an earthquake saturates at the seismogenic width and propagates along strike only, it continues to rupture through a similar stress state, resulting in a constant stress drop that is independent of rupture length. The stress drop to which large events converge is dependent on the stress space available. Compare, for example, the average stress state of models B and H (Figure 3) and the average stress drop (Figure 7). Since t d = f (l), strong faults have higher stress drops than weak faults. [24] From static crack models [Scholz, 1990] the average fault slip is related to the stress drop by u ¼ t s C 2 G ; where t s is the static stress drop and C 2 is a constant that depends on the geometry of the rupture with limits 2/p < C 2 <7p/16, the upper limit for a circular crack and the lower limit for an infinite strike-slip fault in a whole space. For = L (the L model [Scholz, 1982]) and t s constant the dislocation behaves as an expanding crack and slip continues to linearly increase with rupture length until the rupture terminates. For = W (W model) and t s constant, slip is controlled by the fault width and slip is constant as the rupture grows in L. For an elliptical dislocation surface with a nonuniform slip distribution [Yin and Rogers, 1996] the characteristic length is asymptotic. If it is assumed that fault overpressure l limits the amount of slip because of a reduced dynamic stress drop, then the average slip peaks over a smaller in order to maintain a constant t s. Therefore it is assumed that is asymptotic and a function of l(l/w) with the form ð9þ L ¼ 1 þ l L : ð10þ W

9 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING ESE 3-9 Figure 6. t d for different simulations as a function of depth, superposed with equation (8). The events are separated into those that reach the surface and confined events. Figure 7. Model catalogs compiled from simulations under different degrees of pore pressure (Table 1) showing (a) t d and (b) u as a function of rupture length for surface-breaking ruptures. Equations (8) and (11) are shown superposed. The model shows that the stress drop converges to a constant as L grows large because the stress space (Figure 3) is relatively constant along the fault plane. Therefore, as the rupture propagates along strike only, it propagates through a similar stress state and results in the same stress drop for large events. The value to which the stress drop converges depends on extent of the stress space. The average slip is shown to increase with rupture length but at a reduced rate as L grows large, resulting in a smooth transition from L model to W model mechanics at large rupture length.

10 ESE 3-10 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING [25] In Yin and Rogers s [1996] results a slip distribution was imposed over an elliptical dislocation fault surface and the equations were numerically integrated to determine the stress drop. In this model, slip evolves over the rupture surface (see Figure 4, for example), resulting in essentially a nonuniform slip distribution over an approximately elliptical surface, suggesting that is the same for both models. Substituting equation (10) into equation (9), the average slip is then u ¼ C 3t d C 2 G ¼ C 3t d L ; C 2 G 1 þ l L ð11þ W where C 3 = t s /t d is introduced to recast equation (9) in terms of the t d measured in the model. Development of the rupture in the model is controlled by t d, while t s depends only on the final slip distribution over a characteristic length and does not include information about how the slip developed. C 3 is used to recognize this difference between t d and t s. For a step or ramp slipweakening model without overshoot, C 3 =1. [26] The very good quantitative comparisons of equations (8) and (11) with t d and u (Figure 7) from the model show that the inherent complexity is easily quantified. Furthermore, the fit of equation (11) to the average slip (Figure 7b) in all of the model realizations collapses to a single function (Figure 8) such that C 2 Glu C 3 t d W ¼ f l L ¼ l L W W 1 þ l L ; ð12þ W demonstrating that has the proper form for characterizing this model. This form of was used in deriving the expression for determining l for large earthquakes [Miller, 2002], and Figure 8 provides some modeling support for that argument. A form similar to was determined from completely different arguments [Matsu ura and Sato, 1997; Fujii and Matsu ura, 2000], who considered scaling relationships implied by a model of a brittle crust coupled to a deforming asthenosphere under different loading conditions. [27] Equation (11) shows that the average slip is controlled by the fault strength. For strong faults (e.g., small l), slip continues to increase with increasing rupture length, while for weak faults the length dependence falls off quickly as the aspect ratio grows much more than 1. This is directly linked to the rupture process. For a strong fault the high stress drop associated with rupture initiation propagates the instability further along the fault, while the correspondingly low stress drop of a weak fault limits the propagation. In addition, slip increases with rupture length for large events because the high stress drop rupture front feeds back to those parts of the fault sliding at the dynamic friction value. This feedback distance is limited for the low stress drop rupture front of a weak fault. [28] Substituting equation (8) into (11) and using M o = GLWu the seismic moment is M o ¼ 1 C 3 ðc 1 m s m d Þr r gzwl 2 ð1 lþ 2 C 2 1 þ l L ð13þ W The constants C 1, C 2, and C 3 are constrained within tight limits. C 1 is the degree of overshoot of the static failure Figure 8. Collapse of equation (11) used to fit all of the average slip data in Figure 7b to a single function of l(l/w), demonstrating that equation (10) is the appropriate form of for characterizing this model. condition and can be measured in the laboratory, C 2 depends on the rupture geometry and is close to unity, and C 3 is the ratio of t s to t d. The fitting parameters are listed in Table 1. [29] The implication of equation (13) is that there exists a continuous transition from L model [Scholz, 1982] to W model [Romanowicz, 1992] mechanics as the rupture length grows along strike at constant fault width. Equation (13) also predicts that M o / L 4 for small events with L W and l = const, in contradiction to area scaling. However, for a pore pressure profile that parallels the lithostat, as is commonly assumed, l is an increasing function of depth and thus counteracts the length scale contribution of Z in equation (13), resulting in area scaling Comparisons With the Historic Catalog of Strike- Slip Events [30] The earthquake catalog compiled by Wells and Coppersmith [1994] contains hundreds of events from a range of tectonic environments. From this catalog I extracted all events listed with a strike-slip mechanism to compare with the model earthquake catalogs. The database contains most large earthquakes in the modern catalog that are reasonably well constrained, and no attempt was made to filter the data set. Included in this catalog are over 100 earthquakes with strike-slip mechanisms (both left- and right-lateral). The comparison (Figure 9) shows that the different models all fit the data, with average properties that depend on the stress state of the model considered. Because this is displayed as moment versus area, Figure 9 shows that the models predict the average slip of earthquakes over all magnitude ranges. A more detailed study of other catalogs and the implications of equation (13) on earthquake scaling relationships is given by Miller [2002] Surface Displacements [31] Since the model calculates slip along a vertical fault plane in a three-dimensional elastic solid, it is interesting to

11 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING ESE 3-11 Figure 9. Comparison of model catalogs (using equation (13)) of the different model realizations (stars) with the catalog of strike-slip earthquakes (circles) compiled by Wells and Coppersmith [1994]. The agreement between model and observations show that the model successfully predicts the average slip over the range of earthquake magnitudes. investigate the surface response to a large model earthquake to relate to satellite interferometric techniques for inferring and constraining slip distributions at depth. Figure 10 shows the calculated elastic vertical displacement at the surface in response to a large model earthquake. The displacement field is determined by integrating the 33 component of the strain tensor through the depth and assuming zero strain at the base of the model. Large compressional and dilatational zones mark the boundaries of the rupture, and the profile shows complex along-strike variations in the vertical displacements. Since some of this deformation would accumulate as plastic strain over long times, a complex surface profile would result and would provide an additional degree of complexity in generating erosion networks. Because this model can be run over very long timescales, it could be coupled to a model of erosion to study complex geomorphology around active fault zones. 4. Conclusions [32] The purpose of this paper was to show and quantify general properties of a model that includes the dominant Figure 10. Example of the surface displacement field in response to a large model earthquake at depth. The complex surface displacement reflects the complex slip distribution at depth, including large changes that mark the boundary of the rupture and numerous along-strike variations.

12 ESE 3-12 MILLER: LARGE RUPTURES AND INFLUENCE OF FLUIDS ON FAULTING effects of fluids on the earthquake process using a forward model that couples elastic dislocation theory with a simple model for fault zone hydraulics. The assumptions and simplifications allowed the model to be tractable, while keeping the dominant physics intact. Many important processes have been approximated or ignored, but this model shows that many features of the earthquake process can be explored. A more complete treatment of processes currently ignored is discussed by Fitzenz and Miller [2001]. The model demonstrates the simplicity of generating and maintaining complexity along a vertical fault plane by including the dominant mechanical effects of fluids within fault zones. Complexity results because of the extra degree of freedom allowed in stress space by including coseismic changes in fault zone hydraulics. In contrast, complexity in frictiononly models is more difficult to achieve because changes in effective normal stress are not considered and all complexity must arise from highly nonlinear friction behavior. A general result of this model is that the system will achieve a dynamic equilibrium of a high degree of stress state disorder. A further implication of the continuous complexity in the stress state is that there appears to be no lower limit of earthquake triggering because something is always on the verge of failure. Whether or not that grows to a large event depends on the correlated state of stress in the affected region. [33] This general result is consistent with that found in a previous version of the model [Miller et al., 1999], but the previous model did not include dynamic friction and therefore limited the size of an event that could be generated. Including a simple slip-weakening friction model allows the generation and propagation of large model ruptures, and the disordered stress state results in complex slip distributions. Large events converge to an approximately constant stress drop for faults at the same overpressure because the stress state through which large ruptures propagate is similar along strike. A compilation of model catalogs shows that all of the model earthquakes can be quantified by simple relationships for stress drop, slip, and seismic moment. The characteristic length in the model that relates the average slip to the stress drop is found to be asymptotic and a function of the fault overpressure l and the aspect ratio of the rupture. This form of the characteristic length results in scaling relationships for large earthquakes that show a smooth transition from L model to W model mechanics for large earthquakes. These results show a depth dependence of the stress drop for faults at the same overpressure l. 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