Self-organization of elastic moduli in the rock above blind faults

Authors


Abstract

[1] Distributed anelastic deformation accommodates long-term motion in the “lid” region above blind faults. The flowing sequence of cyclic processes may lead to self-organization within lids composed of quartz-rich clay-bearing sedimentary rocks. Coseismic slip on the deep fault imposes strain and displacement on the lid. Sufficient strain brings the lid to frictional failure, producing distributed cracks. Cracked lid becomes increasing elastically compliant over many seismic cycles. The lid tends to evolve so that coseismic displacement barely causes frictional failure; hence, only a few new cracks open within the lid. However, this process cannot increase crack porosity and decrease the shear modulus below values where the material would readily compact under lithostatic pressure. Accumulating sedimentary rocks provide a convenient proxy for this limit. Numerical modeling indicates that the (geodetic) coseismic displacement above a compliant lid is much of the total geodetic displacement that would be observed over a full cycle. Stresses within the lid relax by ductile creep at nearly constant strain with only modest geodetic displacement >100 year after the earthquake for a 1000 year cycle.

1 Introduction

[2] The rock above blind faults accommodates displacement (Figure 1). I use “lid” to refer to this region, as a standard term is not available. The strain within the lid from an individual seismic event need not exceed the elastic limit, but tectonic displacement involves essentially anelastic strains of ~1 in the lid over many earthquake cycles. Ruptures on major crustal earthquakes, including those of the Landers event [e.g., Wald and Heaton, 1994], jumped from the trace of one fault to another at step-over zones. Lids and step-over zones are similar phenomena in that distributed anelastic deformation accommodates long-term tectonic movements. I concentrate on lids above blind faults, as they are geometrically simpler than step-over zones.

Figure 1.

(a) Schematic diagram of compliant damaged zone in the lid of a blind strike-slip fault. In the limit that the zone is very compliant, it accommodates the imposed fault slip between points A and B. (b) The strain directly above a fault with constant slip within an elastic half space is normalized so that the strain at the surface for an infinitely deep fault is 1. The top of the fault is at normalized depth 1. Strain increases slowly with depth and is insensitive to the depth to the bottom of the fault.

[3] Specifically, I examine the interaction above blind faults of co-seismic deformation, pervasive cracking from co-seismic strain, and long-term ductile creep. For geometrical simplicity, I consider a blind strike-slip fault as near Bam Iran [Talebian et al., 2004; Fielding et al., 2009; Poiata et al., 2012] in dimensional, analytic, and numerical examples.

[4] Importantly, observations summarized by Kaneko and Fialko [2011] indicate immediate postseismic creep in the lid does not typically accommodate deep displacement. In addition, shallow fault zones are often more elastically compliant than the surrounding shallow rock [Cochran et al., 2009; Kaneko and Fialko, 2011] (Figure 1). I qualitatively illustrate the tendency for self-organization toward compliance within the lid by starting tectonic activity and the first earthquake cycle with stiff rock (high shear modulus G) extending all the way to the surface. The coseismic elastic strain above the fault increases the shear stress bringing the lid rock to internal frictional failure. Repeated pervasive cracking in each event reduces the shear modulus in the lid to the level into a quasi-steady state where peak coseismic stresses and strains barely produce frictional failure and hence few new cracks. Then, ductile creep accommodates the long-term deformation in the lid.

[5] In addition, the early development of a through-going fault may lead to a compliant lid. The initial shallow and deep slip may occur of a distributed network of faults. Pervasive fracturing from this process may make the lid compliant. Slip then nucleates on a well-defined fault at depth, while lid accommodates slip over a broad region.

[6] In more detail, each deep fault movement imposes displacement boundary conditions and a given strain within the lid. Approach to quasi-steady state is then possible and interesting as long-lived stasis is potentially observable. Each coseismic strain and stress change just brings the lid to frictional failure maintaining the appropriate shear modulus as a function of depth. In this scenario, the lid rock needs to creep in a macroscopically ductile manner in the interseismic interval and relax shear stresses back to their pre-event value without producing more cracks. The geometry of the damaged zone from this process depends on static-stress changes that vary spatially within the lid.

[7] Static damage in the lid is distinct from analogous cases that involve self-organization of damaged rock. Dynamic stresses from strong shaking off-fault damage shallow (10s to 100 s m) regolith [Brune, 2001; Dor et al., 2008, 2009; Girty et al., 2008; Wechsler et al., 2009; McCalpin and Hart, 2003; Replogle, 2011; Sleep, 2011]. (Here “regolith” is the shallow mechanically disrupted region, distinct from chemically weathered soil. Planetary and lunar scientists use the term in this manner [e.g., Blanchette-Guertin et al., 2012]. Blind-fault lids are also distinct from near-fault damage in flower structures from high transient stresses at the fault rupture tip [e.g., Ma, 2008]. Flower structures occur along fault zones that breach the free surface and hence may be distinguished in the field from deformation above blind faults.

[8] I have no applicable field example of self-organization of the shear modulus within the lid of a blind fault. A major purpose of this paper is hence calculations in expectation that the hypothesis can be tested. That is, the shear modulus and, in general, the bulk modulus within lids are reduced below their laterally ambient values. Geodetic studies, in addition to detecting postseismic creep, provide information on elastic moduli. Static seismic slip on nearby faults elastically deforms the surroundings making analysis feasible [e.g., Cochran et al., 2009]. Seismic velocity studies are relevant, but available studies likely lack adequate resolution. Sadeghi et al. [2006] did attempt to resolve shallow structure near the Bam Iran and did not detect low velocity in the lid of the fault. Allam and Ben-Zion [2012] did not attempt to resolve seismic structure in the upper 3 km of southern California.

[9] I present scaling relationships in section 2. In section 3, I show that ductile creep within the lid occurs throughout the interseismic interval unlike frictional creep that occurs mostly in the immediate aftermath of the event. I begin with scaling relationships to obtain the gross stresses and strains involved in the proposed process. I then examine the rheology of shale, as ductile creep is most likely within lids composed of clay-bearing sedimentary rocks. I finally present numerical models of a blind strike-slip fault within sedimentary rocks.

2 Scaling Relationships for Stress and Strain in the Lid

[10] As I have no documented field example, I analyze the geometrically simple case of a buried very long strike-slip fault to illustrate scaling relationships without obscuring them with mathematics. The example is directly applicable to the nearly strike-slip segment of the Bam Iran 2003 earthquake [Talebian et al., 2004; Poiata et al., 2012]. Calculations would be feasible where fault-slip had both strike-slip and thrust components, for example, the 1989 Loma Prieta earthquake [e.g., Wald et al., 1994] and the 2010 Haïti earthquake [Calais et al., 2010; Hayes, 2010]. Step-over zones are even more geometrically complicated, but the gross approach carries through. I begin with the mechanical properties of basin sediments to focus dimensional scaling.

2.1 Ambient Mechanical Properties Within Sedimentary Basins

[11] I consider generic sedimentary basins as in southern California and near Bam Iran to obtain dimensional examples that provide an overview of analytical and numerical results. I begin with ambient properties and the effects of compaction under lithostatic stress. S-wave velocity for stiff sediments is β = 2500 m s-1 from southern California [Allam and Ben-Zion, 2012]. The rock density is ρ ≈ 2200 kg m-3. The shear modulus is

display math(1)

13.75 GPa, retaining insignificant digits. The S-wave velocity is the dominant parameter in (1). I ignore variations in density for brevity.

[12] Accumulating sediments compact under lithostatic load, increasing their shear modulus. I use data on this process as a proxy for the related process where lithostatic pressure closes cracks in damaged rocks. That is, I assume that the interseismic stiffness of damaged rocks at a given depth cannot be less than that of accumulating sediments.

[13] Data are available for accumulating quartz-rich sediments in California. The shear modulus approximately increases linearly with depth in the San Fernando and Santa Clara valleys in California [Magistrale et al., 2000; O'Connell and Turner, 2011]. In general,

display math(2)

where the shear modulus is Gcal at depth Zcal, and it is understood that the shear modulus is less than or equal to that of intact rock. The S-wave velocity of ~600 m s-1 at 50 m depth and the density of ~2200 kg m-3 [Magistrale et al., 2000; O'Connell and Turner, 2011]. Retaining insignificant digits for subsequent use, the bracket term in equation (2) is 15.84 MPa m-1. The shear modulus in equation (2) reaches the shear modulus of intact rocks 13.75 GPa at 868 m. It is likely that the shear modulus of ambient sedimentary rocks continues to increase somewhat below this depth. I do not attempt to model this subtle effect.

[14] Furthermore, the shear modulus of exhumed (originally stiff) sediments that were likely damaged by strong shaking also increases linearly with depth but at a faster rate than for accumulating sediments [Sleep, 2011]. That is, the observed damage did not reduce the shear modulus all the way to that of compacting sediments justifying the use of equation (2) for compacting sediments as a lower limit. For reference, field data constrains the bracket term in equation (2): the S-wave velocity in exhumed sediments at 50 m is ~800 m s-1 for averaged sediments in the San Fernando Valley [Magistrale et al., 2000], ~1200 m s-1 in the Mission Hills anticline in the San Fernando Valley [O'Connell and Turner, 2011], and ~1000 m s-1 within sandstone near Cajon Pass, California [Anderson et al., 2008].

[15] Damage zones around faults are often observed at significant depths. Sediments studied from the SAFOD borehole near the San Andreas Fault at Parkfield at ~3 km depth have S-wave velocity of ~2500 and ~2000 m s-1 in the most damaged domains [Jeppson et al., 2010]. Damage in hard rocks is observed along faults in Southern California below 3 km depth where data are available [Allam and Ben-Zion, 2012]. Damaged rocks have S-wave velocity of ~2500 m s-1, again similar to intact sedimentary rocks, and intact hard rocks have ~4000 m s-1 [Allam and Ben-Zion, 2012]. This velocity limit in equation (2) is reached at 2500 m depth for compacting sediments, indicating that a more sophisticated analysis is required to model deep damage within hard rocks.

[16] Equation (2) constrains whether the damaged zone in the lid can become elastically compliant enough to accommodate much of the displacement on the underlying fault. In the limit of high compliance, the observed displacement across the lid between points A and B (Figure 1) is essentially the slip on the underlying fault. The stresses in the damage zone relax during the interseismic period while total strains and displacements vary slightly.

2.2 Dimensional Calculations

[17] I begin scaling analysis with coseismic elastic deformation and then proceed to interseismic anelastic deformation. In an actual field example, seismologists analyze seismic waves and static displacements to infer the distribution of coseismic slip on a fault plane. Typically, there are patches of high and low slip for well-resolved major events. Patchiness may well continue down to short spatial scales [e.g., Mai and Beroza, 2002]. Fault slip is typically not crack-like with displacement varying smoothly from zero at the top and bottom of the depth range of slip to a maximum in the middle.

[18] The fate of slip patches over many earthquake cycles is not known, but the total slip rate over many earthquake cycles must approach the tectonic rate. Low-slip patches in one event may catch up in subsequent events or creep interseismically. I ignore the latter complication to attend to processes within the lid. In addition, I assume for simplicity that coseismic displacement is uniform with depth from the top to the bottom of the slip zone.

[19] With these simplifications, I assume generic properties for coseismic deformation. The fault plane below some depth ZF = ~1 km slips ΛF = ~1 m, imposing a strain of εF ≈ ΛF/ZF = ~10-3. For this antiplane deformation, the change in shear traction is ΔτF ≈ F. The imposed strain causes a stress change of 13.75 MPa within intact exhumed sediments for the stiffness used in section 2.1.

[20] I begin with the end-member assumption that the lid was unstressed before the event. A single event brings much of the initially stiff lid into frictional failure. To the first order, frictional failure within the lid occurs when

display math(3)

where τ is the shear traction on the fault, μ0 is the first-order coefficient of friction, PN is the effective normal traction on the fracture. The second approximate equality (where g is the acceleration of gravity ~9.8 m s-2, ρ is rock density, ρw is pore-water density, and Z is depth) assumes that the difference between lithostatic stress and fluid pressure acts on the crack. Frictional cracking occurs if the shear modulus is greater than

display math(4)

for an imposed strain. Cracking occurs in each seismic cycle if the shear modulus Gfail is less than the maximum shear modulus for compacting rocks in equation (2). That is, μ0g(ρ − ρw)/εF < Gcal/Zcal. Cracking does not occur if Gfail is greater than the shear modulus of the intact rock.

[21] Equation (4) implies frictional failure in the lid for reasonable coseismic strain ~10-3. I let the coefficient of friction be 1 and use the density 2200 kg m-3 for quick example calculations. Stiff rock (here with an S-wave velocity of 2500 m) is bought to failure by a single event down to a computed depth of 640 m. Pervasively cracked rock with the bracket term in equation (2) of 15.84 MPa m-1 for accumulating sediments fails at strain 0.76 × 10-3. Long term strains ~1 from kilometer-scale slip on the fault cause frictional failure throughout the lid if the stresses do not otherwise relax.

[22] As already noted, there is a tendency for the shear modulus in the lid to self-organize so that the imposed static stress and strain changes just bring the entire lid to frictional failure in each event. Minor damage then statistically restores any cracks that healed in the interseismic interval, leading to a quasi-steady state. In one vertical dimension directly above the buried fault, the S-wave velocity from equations (1) and (4) at coseismic failure is then dimensionally

display math(5)

where it is understood that the relationship cannot be extrapolated to great depths where the computed seismic velocity damaged sediments is greater than the ambient value for intact sediments. The predicted behavior in equation (5) is not strongly dependent on unknown parameters. Three caveats are in order: (1) damage cannot decrease the S-wave velocity below that where lithostatic pressure closes cracks in equation (2). Even loose sand and extremely cracked rock are held down by gravity; they have finite S-wave velocities that cannot go all the way to zero at the free surface. (2) As discussed in Appendix A, rock may have finite frictional strength (cohesion) at zero depth. (3) The damaged zone may narrow with depth (Figure 1) so that strain increases with depth. I address this issue with analytic calculations in section 2.3 and numerical calculations in section 4.3. I address another serious issue in the section 3: equations (4) and (5) presume that interseismic processes in the lid completely relax shear stresses incrementally imposed by deeper fault slip during each seismic cycle.

2.3 Analytical Elastic Solution

[23] As noted in section 2.2, strain near the buried fault tip differs in detail from its overall dimensional value in the lid. The well-known analytical formula for static stresses and strains from a very long strike-slip fault buried within a half space illustrates this issue and to the first order constrains the spatial distribution of static strains [e.g., Jaeger et al., 2007, pp. 425–426].

[24] The model fault at horizontal coordinate x = 0 slips in the perpendicular y-direction. The vertical coordinate z by tradition is negative downward. Symmetric slip of + ΛF occurs on the − x side of the fault and − ΛF on the + x side. This uniform slip occurs between the large depth − z = dbot and the shallow depth − z = dtop. The displacement in the y-direction uy satisfies Laplace's equation,

display math(6)

[25] The zero shear traction boundary condition is ∂ uy/∂ z = 0 at the free surface z = 0. The displacement field is that satisfies equation (6) is then

display math(7)

[26] The slip at the free surface, which is measured in practice, is

display math(8)

[27] The strain at the free surface is

display math(9)

[28] Equation (9) quantifies the features obtained by the dimensional approach. The total displacement is 2ΛF, and the surface strain directly above the fault is

display math(10)

[29] The maximum depth of fault slip dbot term is unimportant if this depth is much greater than the lid thickness dbot > > dtop. This situation is likely for individual blind earthquakes. Over long times, anelastic slip occurs beneath the seismogenic zone so the effective value of dbot is very large and the dbot terms may be ignored in example calculations.

[30] The width of the highly strained zone in the lid (as expected) scales to the depth to the crack tip. Strains above the fault increase gradually with depth and are insensitive to the depth to the base of the fault dbot (Figure 1). The width of the highly strained zone similarly decreases with depth so that the total displacement is conserved. With regard to initial damage from repeated static strains, the analytical model predicts the damaged region extends laterally over a distance comparable to the lid thickness and that the width of damage narrows with depth within the lid. Section 4.3 numerically addresses the geometry of the damaged zone after its elastic modulus has been reduced below that of the initial rock.

[31] The predicted strain becomes infinite at the fault tip implying that the model is inapplicable there. Rather fault displacement decreases continuously upward toward the crack tip so that finite stresses near those for frictional failure are maintained. The two-dimensional numerical models in section 4 represent the fault tip in a more realistic manner. In general, stresses and displacement in the upper part of the lid are not sensitive to the details at the crack tip and on the fault plane (Saint Venant's principle [e.g., Timoshenko and Goodier, 1970, pp. 39–40]), so the shallow part of the analytical solution provides reasonable guidance.

3 Interseismic Creep

[32] Macroscopically creep throughout the lid is required so that very large shear stresses do not build up over many earthquake cycles for macroscopic faults that do not breach the surface. Phenomenologically, tectonic-scale folds above blind thrusts are essentially inelastic. They do not suddenly spring back with hundreds of meter displacements.

[33] I separately consider frictional creep and ductile creep as they produce distinctive interseismic strain and strain histories and ambient stresses. I continue with a blind strike-slip fault so that I do not obscure simple relationships with complicated mathematics.

3.1 Frictional Creep

[34] The well-known construct of rate and state friction provides quantification for frictional creep within a pervasively cracked material. The macroscopic strain is the sum of deformation on a large number of fractures. I consider the fate of a typical fracture to obtain the change in shear stress over the seismic cycle and its history over time.

[35] I apply the strain rate form of a unified theory of rate and state friction compiled by Sleep [1997, 2006] and Sleep et al. [2000], as I am interested in macroscopic distributed strain. That is, failure is macroscopically internal deformation and internal friction but slip on random cracks when viewed in detail. The instantaneous shear traction τ on a crack as a function of normal traction PN is then

display math(11)

where the dominant term as in (3) μ0 ≈ τ/PN represents the approximation that the coefficient of friction has a constant value for a given surface (Amontons’ law), a and b are small ~0.01 dimensionless constants, ε′ ≡ V/W is the strain rate, V is sliding velocity, W is the width of the sliding gouge zone, εref is a reference strain rate, and ψ is the state variable that includes the effects of healing and damage, and the ψnorm is the normalizing value of the state variable, which arises if normal traction changes and with regard to apparent cohesion at low normal tractions (Appendix A). For now, I consider constant normal traction on a given fracture and set ψnorm = 1 without further loss of generality.

[36] Implications of (11) become evident when it is rewritten as a scalar flow law

display math(12)

[37] Note that εref exp(−μ0/a) is effectively a multiplicative constant. Thus, one may set εref to a physically relevant value that fixes μ0 using experimental data. Furthermore, the change of shear stress from the aftermath of the earthquake to the start of the next earthquake is relevant here. One does not need to know μ0 precisely for this purpose. Also, note that the logarithmic singularity at zero strain rate in equation (11) does not arise in equation (12); the predicted strain rate is unobservably small at very low shear stresses.

[38] In general, one needs to know the change in the state variable ψ over the interseismic interval to evaluate equation (12). I use the “aging” evolution law to represent the state variable ψ in terms of the past history of the fault to illustrate well-known implications of equations (11) and (12) [Dieterich, 1979],

display math(13)

where the first term represents healing and the second damage. The variable t is time, εint is the strain to significantly change the properties of the sliding surface, α is a dimensionless parameter that represents the behavior of the surface after a change in normal traction from the work of Linker and Dieterich [1992], and Pref is a reference normal traction. Equation (13) implies that the state variable increases strengthening the surface when sliding is not occurring ε′ = 0. Experiments where P-wave transmissivity directly constrained the logarithm of state and showed that surfaces do strengthen in the predicted manner [Nagata et al., 2012].

[39] Equation (2) empirically represents the tendency of highly elastically compliant material to rapidly anelastically compact at depth and thus not typically be present within the Earth. In terms of equation (13), the state variable increases with time when sliding is not occurring, and the state variable increases, decreasing porosity [Segall and Rice, 1995]. Compaction and closing of numerous cracks increases the shear modulus (Appendix A of Sleep [2011]).

[40] I continue with the history of postseismic stress and strain. I set Pref = PN to compact notation, as (by the assumption of a strike-slip fault) normal traction does not change with a given place in the rock and relegate discussion of this term to Appendix A. The steady-state value of the state variable in equation (13) is then

display math(14)

[41] I consider a simple example to obtain the basic form of the evolution of shear stress and strain to illustrate general features of the seismic cycle. I assume that coseismic slip and shaking do not reset the state variable along many of the fractures in the lid. That is, the state variable is typically large and does not change much in equation (13) over the seismic cycle. The earthquake suddenly increases shear stress by ΔτF. Thereafter, shear stress evolves as

display math(15)

[42] High shear stresses are very quickly relaxed by the rapid strain rates in equation (15). The widely used logarithmic decay of stresses and elastic strains arises as an approximation from equation (15) at large times tpost after the earthquake

display math(16)

where the shear traction is τyield at time tyield. The measurable relative change in stress is independent of ambient normal traction, provided that coseismic strain brings the material to yield.

[43] Alternatively, coseismic damage resets the state variable for some fractures in the lid to a low value ψlow. At long times after the event, the state variable in equation (13) is

display math(17)

when the state variable is still well below its steady-state value in equation (14). The strain rate in equation (12) decreases, as ψ increases in addition to the direct effect from stress relaxation.

[44] Physically, resetting of the state variable to low values by cracking on some fractures is likely to be unimportant to total postseismic strain. Slip on the main fault imposes the geometry of macroscopic shear strain within the lid. Local failure on a crack relaxes the shear stress on that crack. Failure of neighboring cracks over numerous cycles is required to reload the local stress so that the crack can fail again. Thus, the state variable from equation (13) is typically at a high value for the cracks that actually creep. Hence, equation (16) gives a maximum and likely estimate of the observable relative shear stress change over the seismic cycle for frictional rheology.

[45] I obtain the shear modulus and seismic wave velocity at failure in terms of equation (16). These quantities are observable within the lid at quasi-steady state. Coseismic displacement leaves material at frictional failure at μyieldPN, the preseismic stress is PN(μyield − a ln(tcycle/tyield)) ≡ PNμpre, where tcycle is the length of the seismic cycle. The criterion in equation (4) becomes

display math(18)

[46] The S-wave velocity for failure in equation (5) becomes

display math(19)

where the material is compliant enough that observable postseismic slip accommodates all the tectonic deformation. The seismic velocity predicted by equation (19) is much less than that predicted by equation (5) where shear stress fully relaxes in the interseismic interval. The shear modulus in equation (18) cannot decrease below the value where lithostatic pressure closes cracks in equation (2).

[47] It is not necessarily to precisely know physical parameters to obtain the implications of equations (16), (18), and (19). In addition, equation (18) applies for both frictional and ductile relaxation of stress when μpre is defined more generally to give pre-event shear stress, which remains positive. If μpre and hence the preseismic stress in the lid are small, a small coseismic strain does not bring the lid to frictional failure. The coseismic deformation of the lid is elastic, and the interseismic deformation is ductile. Conversely, a large coseismic strain brings the lid to frictional failure even if the preseismic stress is ~0.

[48] With regard to geodetic methods, comparison of immediate postseismic data with preseismic data provides coseismic slip and the shallow slip deficit compared with the deep fault. One then extrapolates observed postseismic creep measured over the remaining seismic cycle. The definition of postseismic, however, depends on available observations. In an optimal case, seismologists might have an array of strong-motion instruments around the fault. Off-fault deformations might become observable (defining tyield) after the passage of the crack tip in a fraction of a second. More likely, the duration of slip over the entire fault ~30 s for Landers [Wald and Heaton, 1994] is relevant if there are nearby continuous geodetic instruments. I let the seismic cycle time tcycle = 1000 a (3.15 × 1010 s), for an example. The argument of the logarithm in equation (16) is then between 1010 and 1012. Much of the eventual displacement occurs in the first year 3 × 107 s, 6/10 and 2/3 of the total, respectively. The eventual displacement is well constrained if continuous geodetic measurement are available, as more than half of it occurs within the first year. The computed final postseismic slip is often insufficient to balance slip on the underlying fault [e.g., Kaneko and Fialko, 2011].

[49] Overall, neither μyield nor a are precisely known from actual field examples, but this information is not necessary to obtain the essence of frictional behavior. The apparent coefficient of friction of 0.6 observed within boreholes in stable regions is a possibility [Townend and Zoback, 2000]. It is not likely to greatly exceed 1 for cracked rock. I use generic parameters of a = 0.01 and μyield = 1 as an example; 23% and 28% of the peak stress relaxes before the next event, respectively for my two time ratios.

3.2 Ductile Creep

[50] I contrast the time history implied by ductile creep with that implied by frictional creep in section 3.1. Ductile rheology is attractive for exhumed quartz-rich clay-bearing sedimentary rocks. I begin with the scalar linear Maxwell viscoelastic rheology, which leads to gradual relaxation of postseismic shear stress. The strain rate is the sum of the effects elasticity and viscous creep

display math(20)

where η is viscosity.

[51] The shear stress at constant strain is approximately applicable with a very elastically compliant damaged lid (Figure 1) and evolves as, τ = τpost[1 − exp(−Gt/η)], where the postseismic shear stress is τpost. The well-known Maxwell time tMax = η/G illustrates the effect of viscous creep. If this time is much longer that the earthquake cycle time, negligible viscous creep occurs; computation proceeds with elastic and frictional formalism. If the Maxwell time is much shorter than the cycle time, shear stresses quickly approach zero, and the net effect as assumed in equations (4) and (5) is that the lid is not under shear stress at the time of the next earthquake.

[52] The main difference with frictional rheology in section 3.1 is that the strain rate decreases slowly over the interseismic interval rather than very rapidly at first. I give two examples for a cycle time of 1000 a with the intact S-wave velocity of 2500 m s-1 and the cracked rock velocity of 600 m s-1. I retain the sediment density of 2200 kg m-3. The computed viscosity with this Maxwell time is 4.3 × 1020 and 2.5 × 1019 Pa s, respectively.

[53] The long-term anelastic strain over the geological time tgeol to form a tectonic feature, tgeolτ/η, provides field constraints on the order of magnitude of viscosity. I give a generic example applicable to lids of blind faults in California, Algeria, Haïti, and Iran. A strain of ~1 occurs over ~1 million years. The normal traction at ~1 km depth is 22 MPa, where faults and ductile deformation coexist, providing a scaling for stress. (A similar stress is needed to lift a protruding 1-km anticline against gravity.) The computed viscosity is 6.9 × 1020 Pa s, similar to that obtained from considering Maxwell times for stiff rock.

[54] Conversely, very low sediment viscosities cannot extend to depths of even a few 100 m within regions with significant topographic relief. Otherwise, the relief would have already spread from gravitational forces. I provide an example for shale outcropping on the Niagara escarpment [Barlow, 2002]. This rock has been exposed at the base of a cliff for ~104 a; the inelastic strain again is ~1. The shale is buried about 100 m deep, so the shear stress is ~1/10 that in the previous example. So the long-term viscosity is ~6.9 × 1017 Pa s, which would imply a brief Maxwell time even in compliant rocks. This material would have a strain of 1000 for the stress of 22 MPa over 1 Ma in the previous paragraph compared with observed strain of ~1 involved with producing observed topographic relief. Material with this viscosity would have spread laterally rather than forming an anticlinal hill. I discuss more sophisticated rheologies for interbedded clay-bearing sand-rich rocks in Appendix A.

3.3 Compactional Creep and Postseismic Subsidence

[55] Postseismic subsidence is a way to detect distributed co-seismic brittle failure on cracks. Brittle fracture dilates material reducing the value of the state variable on the failed cracks. This dilation reduces the postseismic low-amplitude shear wave velocity. It also produces co-seismic uplift and postseismic subsidence. Fielding et al. [2009] studied the latter near Bam Iran.

[56] I quantify my analysis by considering an empirical relationship between state and porosity f,

display math(21)

where ϕ is a reference porosity where ψ = 1 and Cε is a dimensionless coefficient [Segall and Rice, 1995]. I assume that cracking reduces the state variable within numerous fractures to small values immediately after the earthquake. The state variable in the rest of the rock mass does not change significantly. Combining equations (17) and (21) yields that porosity within failed cracks decreases with the logarithm of time

display math(22)

where t is time after the earthquake and porosity change is measured after t0. The dilatation occurs over a depth scaling with the lid thickness ZF. So the initial uplift and subsequence subsidence with quasi-steady cycles are approximately this porosity change times the lid thickness, ΔU ≈ ΔfZF.

[57] The construct dilatancy angle (the ratio of dilatational strain to anelastic shear strain Φ) in engineering geology [e.g., Ribacchi, 2000; Alejano and Alonso, 2005; Zhou and Cai, 2010] allows estimation of the coseismic change in porosity. The observed Φ ratio in jointed sedimentary rocks at normal tractions relevant to lids is a few tenths. That, is Δf ≈ ΦεF and ΔU ≈ ΦZFεF.

[58] In practice, geodetic studies measure the postseismic subsidence ΔUobs for a finite interval after the earthquake, and one must infer the total subsidence for the earthquake cycle ΔU. I base an example calculation on the work of Fielding et al. [2009] for Bam, Iran. The coseismic rupture was complicated enough that they could study postseismic subsidence but not directly extract the coseismic uplift related to dilatation. Their first measurement was 12 days after the earthquake, and they continued to measure for ~3.5 years. Subsidence of ~0.02 m apparently following equation (22) occurred over that time. The unmeasured subsidence before and after the event does not sensitively depend on those times. I assume 30 s for the beginning and 1000 years for the end. These times imply unrecorded early subsidence of 0.046 and 0.024 m of future subsidence for a total of 0.09 m. The brittle anelastic shear displacement associated with this subsidence is 0.09 m divided by the dilatancy coefficient. For example, Φ = 0.3 implies ~0.3 m of anelastic shear displacement. Taking this result at face value, the anelastic slip was a significant but not dominant part of the slip per cycle of ~1 m. Ductile shear deformation could well accommodate the remaining slip over the next ~1000 years.

[59] This behavior differs from that of ductile compaction and poroelastic compaction. The difference between lithostatic and pore pressure drives both ductile compaction and brittle compaction. However, the ductile creep rate depends on this pressure linearly or to a small power. As with ductile shear creep, deformation spreads out over the interseismic interval and is not particularly fast just after the earthquake. Ignoring ductile creep is thus justified in my analysis and the study by Fielding et al. [2009].

[60] Some caveats are in order with regard to poroelastic effects. Fielding et al. [2009] did not have data until 12 days after the event to fully resolve that equation (22) applies. Pore pressure declines exponentially over a time scale related to permeability, pore compliance, and the length scale of pressure variation. In that case, the observed subsidence of 0.02 m could well represent most of the total subsidence per cycle and not require significant brittle dilatation. Unmodeled poroelastic effects would reduce our computed brittle subsidence per cycle of 0.09 m and the coseismic brittle shear deformation of 0.3 m. This would increase the shear deformation available for ductile creep in the lid.

4 Numerical Calculations

[61] I present one-dimensional spring-slider models to illustrate cyclic time dependence. Two-dimensional models of a very long strike-slip fault illustrate the long-term evolution and geometry of damage. These models address the complications that relaxation of stress in one place may increase the stress in another and that spatial variations in elastic moduli affect stress and strain.

4.1 One-dimensional Models

[62] Spring-slider models illustrate the effect of ductile rheology on time-dependent creep. Movement on the buried fault imposes a strain εF each earthquake cycle of duration tcycle. The shear stress is the elastic strain times the shear modulus. The creep rate is a function of this stress. Stress decreases in the interseismic interval as

display math(23)

where the anelastic strain rate ε′(τ) is a function of the shear stress τ. This function collects brittle and ductile effects, it is ε′(τ) = τ/η for linear viscosity and ε′(τ) = (τ − τY)/η where τY is the yield stress for a Bingham fluid (Appendix A). The strain rate is then zero when τ ≤ τY. In both cases, the ratio of the cycle time and the Maxwell time η/G determine behavior.

[63] Time-dependent calculations were iterated until cycles came into quasi-steady state. Figure 2 shows the stress history for linear viscosity and a Bingham fluid calibrated to have the same stresses at the beginning and end of a cycle. The behavior of a Bingham fluid with a lower viscosity is also shown. The curves differ somewhat, but in all cases, anelastic strain is spread out over the earthquake cycle as expected.

Figure 2.

One-dimensional models of shear traction were computed for a large number of seismic cycles until steady behavior was obtained. Time is normalized to the length of the seismic cycle and shear stress to the Coulomb ratio. There is no frictional creep in the model. Material parameters are adjusted to give reasonable variations of the Coulomb ratio. The linear viscosity model and Bingham model A have similar Coulomb stress ratios at the ends of the cycle and could not likely be distinguished using real data. Even the curvature of the linear viscosity line is not obvious. Bingham model B has a lower viscosity than model A and lower overall stresses. The Coulomb stress ratio increases by 0.232 in each seismic event. The Bingham yield stress is at a Coulomb ratio of 0.44. The Maxwell time normalized to the cycle time is 0.375 in the linear model. The normalized Maxwell time defined from ηshale in (A1) is 0.09375 and 0.0402 retaining extra digits in Bingham models A and B.

4.2 Two-dimensional Antiplane Numerical Formulation

[64] A long vertical strike slip fault is convenient as the equations of motion are simply solved. It is also convenient in that material advects parallel to the fault plane and the free surface and hence remains over geological time at a constant position in the modeled vertical plane of the fault and the free surface. My ideal strike-slip fault does not produce topography, so it is unnecessary to self-consistently include stresses from topographic edifices in the formulation. It is also unnecessary to consider erosion, deposition, and the behavior of uplifted and downwarped rocks.

[65] I model the time-dependent problem where the fault lid can self-organize toward quasi-steady cycles. Numerical methods, which are discussed in Appendix B, determine the horizontal displacement uy in the y (north for discussion purposes) direction of slip, the stress field, and anelastic strain. The vertical coordinate z is positive upward, and the horizontal component x is perpendicular to the fault plane. The nonzero components of the stress tensor depend on the x and vertical z derivatives of uy, in terms of simple shear and in the absence of anelastic strain,

display math(24)

[66] Including anelastic strains ε, the stresses are

display math(25)

[67] Ignoring inertia (as I am interested in inter-seismic deformation), the stress equilibrium equation is

display math(26)

[68] For a nonlinear material, the anelastic deformation rate depends on the second invariant of deviatoric stress math formula, normalized so it gives shear traction in simple shear. The Drucker and Prager [1952] approximation uses this invariant and the mean normal stress (for simplicity here, the lithostatic stress) to define rheology. The rate of anelastic strain, for example, for the x-component is then formally

display math(27)

where ε′ is a well-behaved function of the stress invariant that can include both ductile and frictional effects.

[69] For present purposes, frictional creep is likely to occur along cracks and hence in parallel with distributed ductile creep in clay-bearing rocks. The creep equation at constant state then becomes in analogy with equation (12)

display math(28)

where the normal traction PN is for simplicity the lithostatic stress, εref collects the state variable term, and the other physical parameters in general may vary spatially and with time.

[70] I consider damage that reduces the shear modulus so that the lid above the blind fault can approach quasi-steady cycles. Physically, frictional creep involves cracks and hence damage. Ductile creep is much less likely to open cracks. A simple formulation for damage arises by assuming that new cracks increase the compliance 1/G linearly with respect to the frictional part of the strain rate in equation (28). A simple form is

display math(29)

where GD has dimensions of shear modulus (stress). This formulation is stable in that compliance is always positive. Equation (29) ignores gradual compaction and stiffening of the material between earthquakes. This complication and the effects of porosity change on stress should be included in models of field examples. The limit in equation (2) takes compaction under lithostatic stress into account but in a very simple matter.

[71] The actual value of the damage coefficient GD is not obvious. The ratio Ginit/GD needs to be large enough that the ~1 strains in the lid after earthquakes have some relevant effect over geological time. Shear strain is likely to lead to significant dilatancy. The ratio of dilatant strain to shear strain during initial sliding at low normal tractions applicable to the shallow subsurface (dilatancy angle in engineering geology [e.g., Ribacchi, 2000; Alejano and Alonso, 2005; Zhou and Cai, 2010]) is a modest fraction of 1. As I am aware of no data that constrain Ginit/GD, I opt for numerical convenience and set the parameter to the arbitrary and perhaps high value 180 so that coseismic strain in the first few cycles significantly reduces the shear modulus in the lid. The relative change in shear modulus during any one cycle is still small and decreases as damage builds up toward slowly varying conditions that represent a realistic mature lid. This quasi-steady state is the objective of the calculations.

4.3 Two-dimensional Model Parameters

[72] Scaling relationships permit a tractable number of numerical models. From equation (18), the factor (ρ − ρf)μ0/ΛF facilitates considering fluid pressure. My models may be rescaled to keep that factor constant by reducing fault slip ΛF. The rescaled stresses then are the computed stresses times (ρ − ρF)/ρ, for example, to account for hydrostatic fluid pressure keeping rock density constant. The ratio of the seismic cycle time to the Maxwell viscous relaxation time determines whether ductile creep is important. I represent fluidity 1/η as a function of depth alone in the natural unit of 1/Ginittcyl. Stresses relax on the scale of one seismic cycle if the fluidity is 1 in this unit. Models where this unit is much less than 1 are essentially elastic and brittle, and the viscosity is not relevant. Stress in models where this unit much less than one quickly relax and are incompatible with the persistence of topographic relief.

[73] I assume reasonable material properties for the lid and let the entire domain have these properties. The density of sedimentary rocks is 2200 kg m-3; I assume μ0 in equation (28) is 0.8 and no fluid pressure. The rate and state parameter a = 0.01 determine the range in the coefficient of friction over the seismic cycle in the absence of viscous creep.

[74] The fluidity is constant in the region of interest between the surface and 2 km depth and decreases linearly to zero at 3 km depth to represent hard upper crustal sediments. It is zero until 15 km depth and then increases linearly to the base of the model to represent deep ductile crust. This basal viscous layer had very little effect on the results for the lid.

[75] Strain scales as the ratio of fault slip to the depth to the top of the blind fault which varies between 1 and 3 km in various models. I vary depth to the model fault tip and keep coseismic displacement and the cycle time constant. The model fault slips a total of 2 m every thousand years, implying a reasonable long-term rate of 2 mm a-1. The model is antisymmetric about the fault plane, so the fault material on the model side of the domain slips 1 m below some depth. (For example, 2 km depth implies that the numerical node at 1950 m does not slip, and the one at 2050 m does slip.)

[76] The remaining model features allow calculations to tractably focus on the lid. The blind fault suddenly begins to slide at its full geological rate as the situation after numerous cycles is of interest. Artificial basal and lateral boundary conditions are far removed from the lid (Appendix B). Pristine lid material with Ginit=13.75 MPa continues downward as in a deep sedimentary basin but unphysically to great depth at the 19.95 km base of the model.

[77] I make no effort to accurately represent the rupture part of the seismic cycle as in the models of Kaneko and Fialko [2011] and keep fault displacement constant with depth. Thus, there is no deep fault creep in the models that would obscure the effects of creep within the lid. The computed quasi static stress immediately after the earthquake is the sum of the preexisting stress and stress from elastic response from imposed movement on the fault. Typically, the computed stress at numerous numerical nodes is high enough that the strain rate in equation (28) is impossibly large. Rock in the Earth fails coseismically before such stress levels can be reached. The model computes frictional strain from the strain rate in equation (28) initially using unphysically small (e.g., 10-30 s) time steps. After a tractable number of steps, the shear stress is everywhere low enough that the strain rate implied by equation (28) is physically reasonable for a quasi static calculation and the stress field and coseismic anelastic strain are physically possible for a real fault zone. In particular, computed anelastic strain smoothens the imposed singularity at the fault tip. The postseismic time in my plots begins at 10-6 a (~30 s) after fault slip. Negligible viscous deformation in equation (28) occurs within 30 s after rupture.

4.4 Two-dimensional Model Results

[78] I vary the depth to the fault tip to illustrate basic properties of the approach of the shear modulus to the linear relationship with depth in equation (2). Damage occurs only at shallow depth when the fault-tip is 3 km deep (Figure 3). The transition to behavior in equation (2) occurs with the fault-tip at 2 km depth. Stresses from coseismic displacements relax in the interseismic period for model 3 F-2Z with normalized fluidity of 3/Ginittcyl. Subsequent coseismic slip increases stresses at shallow depths causing distributed frictional failure. Preseismic stresses are higher in model 1 F-2Z with normalized fluidity of 1. Damage approaches the limit in equation (2) in model 3 F-Z1 with 1 km fault-tip depth. The damage zone is wider in the model with the deeper fault-tip, and the preseismic stresses are lower in the higher fluidity model (Figure 4).

Figure 3.

Computed shear modulus above the fault for models after 50 cycles. For example, model F3-Z1 has normalized fluidity 3 and depth to fault tip of 1 km. The shear modulus cannot be less than that of accumulating sediments nor greater than that of intact rock. The shear modulus is not evaluated at depths shallower than 50 m. The shear modulus at very shallow depths would decrease toward the accumulating sediment limit with more model cycles.

Figure 4.

Computed shear modulus in GPa and preseismic shear stress in MPa for models (a) 1 F-1.5Z and (b) 3 F-1Z. As expected, the damaged zone is narrower for model 3 F-1Z with the shallower crack tip. Shallow stresses are close to frictional failure in model 1 F-1.5Z with a normalized fluidity of 1 and much lower in model 3 F-1Z with a normalized fluidity of 3. Note that the lateral edge of the damaged zone is subvertical. The 13.74 GPa contour marks the edge of damage. Note that the free surface boundary condition is imposed between 50 m depth and a phantom point 50 m above the surface. Shear modulus and shear stress are thus not evaluated above 50 m depth, and shallower contours are not shown.

[79] Figure 5 illustrates the evolution of displacement at the free surface as would be measured by geodetic methods. Significant coseismic displacement occurs, which is accommodated by anelastic and elastic strain within the lid. Stresses within the damaged zone relax in the interseismic period by ductile creep. Displacement from ductile creep becomes evident ~100 a after the earthquake. Ductile creep produces some (~0.15 m) of the long-term displacement of 1 m per cycle over 7 km away from the fault.

Figure 5.

Computed displacement relative to point directly above the fault at the free surface as a function of time for models F3-Z1 and F1-1.5Z at various distances from the fault. The full displacement per cycle of 1 m is not reached 7 km from the fault. Ductile creep becomes evident at long times.

[80] A significant part of the anelastic strain in the models presented so far is coseismic and frictional, as inferred for Bam Iran. Fault slip imposes a strain, and the stress from this strain is proportional to the shear modulus. The postseismic stress at very shallow levels in all the models would decrease with further iterations as the very shallow shear modulus evolves to the limit in equation (2). However, the model strains for rupture tips at 1 and 1.5 km depth are large enough to require frictional failure in the lid, even if preseismic stress is zero.

4.5 Wide Compliant Lid Zone

[81] The faults in the models began slipping on a deep well-defined plane. Alternatively, initial deep slip may have been distributed before it nucleated onto a plane. Distributed blind slip likely caused distributed shallow damage. In addition, shallow damage leads in rock–water reaction that likely decreases the shear modulus [Girty et al., 2008; Replogle, 2011].

[82] These processes are not easily constrained. With forethought, I present model W to illustrate their net effect (Figure 6). The fault tip is at 3 km depth. The normalized fluidity is 6, so preseismic stresses are low. Coseismic stresses produce very shallow damage away from the imposed damaged zone but not deeper.

Figure 6.

Computed shear modulus in GPa and pre-seismic shear stress for model W after 12 cycles as in Figure 4. The damaged zone changes little from its imposed state except at very shallow depth away from the fault. Preseismic stresses decreased to low levels.

[83] Strains within compliant damaged zone are imposed by its stiffer surroundings. Much of the displacement is coseismic, distributed, and elastic (Figure 7). Stresses within the compliant zone relax but produce only small displacements. The Maxwell time in deep stiff rocks is shorter than the Maxwell time in shallow rocks. There are small (~0.05 m) antithetic displacements before somewhat larger displacements accommodate part of the long-term fault slip. I am aware of no example were such antithetic displacements have been observed.

Figure 7.

The displacement history for model W as in Figure 5. There are small antithetic displacements at intermediate times.

5 Discussion and Conclusions

[84] Distributed deformation accommodates motions in the lids above blind faults and within step-over zones. Self-organization is possible, especially within lids composed of quartz-rich sedimentary rocks where clay-bearing beds may deform by ductile creep. Coseismic slip on the deep fault imposes essentially strain boundary conditions on the lid and stresses proportional to the shear modulus G. Stress relaxes in the interseismic interval on the Maxwell time scale of η/G. Each coseismic frictional failure produces cracks that make the lid more elastically compliant. The postseismic stress within a sufficiently compliant lid will barely produce frictional failure and hence few new cracks form. The shear modulus tends to stabilize at this condition, where few new cracks form in each seismic cycle. In general, quasi-stasis is long lasting and hence potentially observable.

[85] The stress relaxation part of this process requires a limited viscosity within the lid such that the Maxwell time scale is comparable to the seismic recurrence time. Estimates of the effective viscosity from tectonic stresses and strain rates do fall with the range. Much lower viscosities are precluded by the persistence of topographic relief. Much higher viscosities preclude significant ductile creep.

[86] The shear modulus within lids is measurable by seismic and geodetic methods. The expected region of compliant rock is relatively shallow, less than 1 km. Lithostatic stress compacts crack-like porosity, so coseismic cracking cannot reduce the shear modulus to arbitrarily low levels. I used the observed shear modulus within accumulating sediments in equation (2) as a proxy for this lower limit.

[87] Numerical models illustrate potentially observable postseismic behavior around lids. With an actual earthquake, modelers would know the coseismic slip distribution on the fault plane. They would have some constraint on the earthquake cycle time. They might know the surface slip in previous events. They would have little information on slip at depth in previous events. Once the shear modulus had been mapped within a lid, elastic calculations using the slip on the underlying fault yield stress changes within the lid. Time-dependent modeling of the interseismic period yields pre-event stresses and where coseismic slip brought the lid to frictional failure.

[88] The models illustrate the behavior of an elastically compliant lid as observed by geodetic methods. The lid deforms elastically in the coseismic period; elastic strains accommodate much of the underlying fault slip. The stresses within the compliant lid relax at nearly constant strain over the interseismic period. Only modest geodetic displacement occurs during the 100-year period after the earthquake for a 1000-year cycle.

Appendix A: Rheology of Shale and Bedded Sediments

[89] The effective rheology of shale is complicated and difficult to measure in the laboratory as it is hard to duplicate in situ conditions and geological time scales. Industrial hydrofracture to tap gas and petroleum deposits involves sudden stress changes and is not necessarily a good analog for long-term creep. I discuss two issues: (1) interbedded shale and sandstone may well have Bingham viscous rheology; and (2) it is difficult to tell whether shale has cohesion, that is, finite shear strength a zero normal traction.

[90] Macroscopic strains occur within interbedded shale and sandstone in field-scale folds. Shale beds deform from shear traction on bedding planes. This process allows flexural slip between sandstone beds and hence anisotropy. I consider the simpler case where shale beds cannot geometrically accommodate all the macroscopic strain and anelastic deformation occurs equally within both sandstone and shale.

[91] The net effect may be that the sediment mass is a Bingham Fluid. Macroscopic shear stress partly is supported by sandstone and partly by shale. The shale creeps so that shear stresses relax if the sandstone does not deform. At low shear stresses, the sandstone supports macroscopic stresses that are too low to cause its frictional failure; the macroscopic creep rate is zero. At a somewhat higher shear stress, the sandstone creeps in friction at a constant stress, and the shale supports the remaining macroscopic stress. The shale and hence the rock mass creep in a ductile manner with a strain rate proportional to the shear stress supported by the shale. The macroscopic creep rate is

display math(A1)

where sandstone fails in friction at macroscopic stress τsand and the macroscopic viscosity from shale creep is ηshale. At a still higher macroscopic stress, the shale as well as the sandstone may fail in friction. I provide spring-slider examples with a Bingham fluid in section 4.1.

[92] Next, it is hard to demonstrate true cohesion in shale because it is hard to do laboratory experiments at very low normal tractions. Figure A1 shows the failure (starting friction) stress for shale measured by Yang and Cheng [2011]. The points reasonably fit by a straight line implying finite cohesion at zero normal traction.

Figure A1.

Observed shear stress for frictional failure of the Longtan shale, data from Figure 1 in the work of Yang and Cheng [2011]. A straight (dashed) line extrapolating to finite shear traction at zero normal traction provides a reasonable fit to the data. The Linker and Dieterich [1992] relationship (A3) also provides a fit (solid curve) that extrapolates to the origin. Its parameters are α = 0.30, μref = 1.07, and Pref = 2.8 MPa.

[93] However, rate and state friction predicts similar behavior with true cohesion. Mathematically, the steady state coefficient of friction implied by equations (11) and (13) should be independent of normal traction. This feature determines

display math(A2)

and here allows extrapolation of starting friction data on intact rock to low normal tractions.

[94] I consider intact shale where the state variable has a given value. I define failure and starting friction as deformation at a given strain rate. It is not necessary to know the state variable and the failure velocity if I consider the actually measured the change in starting friction with respect to normal traction. From (A2) and (11), it is

display math(A3)

where the starting coefficient of friction is μref at PN = Pref. This curve also provides a reasonable fit to the data in Figure A1. Two (τ, PN) points mathematically determine the linear fit and the fit from (A3), so both curves have the same degrees of degrees of freedom. It is unproductive to use statistics to see which curve fits better given the small number of data points.

Appendix B: Numerical Methods

[95] Numerical solution of the equation of motion (26) on a square grid with dimension Δx is straightforward. I consider a node with displacement in the y (north) direction of u0. The surrounding node displacements are uup, udown, ueast, and uwest. The vertical coordinate z is positive upward, and the horizontal component x is positive to the west. I obtain stresses from strains. For an elastic material, the stresses are τup = Gup(uup − u0)/ΔX, τwest = Gwest(uwest − u0)/ΔX, τdown = Gdown(u0 − udown)/ΔX, and τeast = Geast(u0 − ueast)/ΔX. It would be straightforward but cumbersome to include anisotropy from bedded flat-lying sedimentary rocks by making the value of G in the horizontal derivative differ from the vertical one.

[96] Anelastic deformation in (28) is easily included. Then, the stress on (for example) the up side of the node is

display math(B1)

where Aup ≡ εz(up)Δx is the past anelastic displacement between these 2 nodes. With these anelastic displacements, the equilibrium equation at this node becomes

display math(B2)

[97] For convenience, I define the shear modulus at the displacement nodes and obtain the shear moduli between nodes in (B2) by linear averaging.

[98] The rate of anelastic displacement in (28) depends conveniently on the stress between the nodes. The anelastic strain rate for example above the node is

display math(B3)

where Δt is the numerical time step and εref is 1 a-1 in the models. The normal traction is the lithostatic pressure and fluid pressure is assumed to be zero for simplicity. I obtain the stress invariant at displacement nodes from the shear modulus at that point and the elastic displacements between this node and the vertical and horizontal neighboring nodes. I evaluate the exponential in (B3) at each displacement node and then average (here with the up node) to evaluate (B3).

[99] The time-dependent problem is easily solved by alternating between the balance of force from elasticity (B2) and anelastic creep (B3). I obtain new displacements from given boundary conditions and previous anelastic deformation from (B2) by successive over-relaxation. The anelastic deformation between all node pairs is then augmented in a time step. This main restriction is that this numerical time step needs to be small enough that the additional displacements only slightly change the stress field. This condition requires very short time steps if there are high frictional creep rates in (B3). Still this procedure is tractable in that the brief displacements add up to finite amounts that reduce stresses and hence frictional strain rates. That is, the initial very rapid frictional creep in the model is effectively coseismic. So it is not necessary to precisely specify the spatial distribution of earthquake slip on the fault plane from a dynamic rupture.

[100] I select boundary conditions that are convenient for modeling the lid and pay little attention to the details at great depth or horizontally well away from the fault. The natural boundary of free slip ∂ u/∂ z = 0 exists at the free surface. This condition is easily applied by having the uppermost physical nodes a half grid spacing down in the Earth and a phantom layer with the same displacements one-half grid spacing up in the air. The zero stress and displacement between the two layers implies that there is no horizontal shear traction or horizontal motion between the layers. It is thus not necessary to know physical properties at the free surface and the singularity in the friction equation (B3) at the free surface does not arise. The natural antisymmetry boundary condition u = 0 exists in the lid directly above the fault. The displacement on the fault plane is incremented a given amount during each earthquake and remains constant in the interseismic interval. That the displacement at artificial lateral boundary is incremented each time step by ΔtVF where VF is the half rate of long-term slip on the fault.

[101] For physical reasonability, I include intraplate stress to have fault movement effectively driven from the side. I begin calculations immediately after an earthquake with 2 m displacement on the lateral boundary and 1 m displacement on the fault. Displacement between the seismic fault and the lateral boundary fluctuates between 1 m and 2 m during the earthquake cycle but does not systematically increase over time. That is, the macroscopic stress drop is a reasonable value of ~50%. Intraplate stress has a minor effect on the behavior of the lid. Free-slip is a tractable bottom boundary condition that does not significantly affect the lid region as long as the base of the domain is much deeper than the base of the lid.

[102] Equation (29) represents damage that reduces the shear modulus. Physical restrictions exist on changing the shear modulus. First, the total effect of anelastic creep and change in the shear modulus cannot increase the stress. This precludes healing (not modeled in this paper) that rapidly increases the shear modulus. Next, creep dissipates elastic strain energy and thus cannot increase that quantity. The elastic strain energy from the elastic strain math formula cannot increase during the creep and damage part of the time step where the displacements remain fixed. This criterion is satisfied as creep reduces the elastic strain and damage decreases the shear modulus. I apply damage in parallel with creep using results of solving (B2). In general, damage from cracks dilates the material and causes in-plane displacements. I ignore this effect for simplicity on the ground that a small density of cracks suffices to reduce the shear modulus and that cracking partly converts equidimensional pores into cracks in a sand-rich sedimentary rock without changing the bulk density. My discussion of postseismic subsidence in section 3.3 assumed that porosity reached quasi-steady state.

[103] I use a square grid spacing of 100 m and make the domain large compared to the lid region to 19.95 km depth and laterally to 40 km. The 100-m spacing precludes resolving the fine details of displacement; a very fine grid is numerically intractable.

[104] I extend sedimentary properties to depth for numerical convenience. I start with a uniform elastic material with Ginit=13.75 MPa. I retain the density of 2200 kg m-3 from earlier examples and extend it to depth. I let μ0 in (28) be 0.8 and the rate and state parameter a = 0.01. I represent fluidity 1/η as a function of depth alone in the natural unit of 1/Ginittcyl. Stresses relax on the scale of one seismic cycle if the fluidity is 1 in this unit. The fluidity is constant between the surface and 2 km depth and decreases linearly to zero at 3 km depth. It is zero until 15 km depth and then increases linearly to the base of the model. Physically, the deep crust below the seismogenic zone likely deforms in a ductile manner. The free-slip basal boundary condition and this ductile layer preclude gradual model deformation associated with deep processes so that the models isolate effects associated with the lid.

Acknowledgments

[105] Yuri Fialko promptly answered several questions. Amir Allam and Yehuda Ben-Zion provided an advance copy of their seismic results. Two anonymous reviewers provided helpful comments. This research was in part supported by NSF grant EAR-0909319. The latter grant is funded under the American Recovery and Reinvestment Act of 2009 (ARRA) (Public Law 111–5). This research was supported by the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. The SCEC contribution number for this paper is 1693. The 2011 SCEC meeting indicated the possibility of self-organizing processes above blind faults.