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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, B10308, doi:10.1029/2006JB004396, 2006 Click Here for Full Article Effective normal stress alteration due to pore pressure changes induced by dynamic slip propagation on a plane between dissimilar materials John W. Rudnicki1 and James R. Rice2 Received 16 March 2006; accepted 11 July 2006; published 28 October 2006. [1] Recent, detailed examinations of fault zones show that walls of faults are often bordered by materials that are different from each other and from the more uniform material farther away. In addition, they show that the ultracataclastic core of mature fault zones, where slip is concentrated, is less permeable to flow across it than the adjoining material of the damage zone. Inhomogeneous slip at the interface between materials with different poroelastic properties and permeabilities causes a change in pore pressure there. Because slip causes compression on one side of the fault wall and extension on the other, the pore pressure on the fault increases substantially when the compressed side is significantly more permeable and decreases when, instead, the extended side is more permeable. This change in pore pressure alters the effective normal stress on the slip plane in a way that is analogous to the normal stress alteration in sliding between elastically dissimilar solids. The magnitude of the effect due to induced pore pressure can be comparable to or larger than that induced by sliding between elastic solids with a dissimilarity of properties consistent with seismic observations. The induced pore pressure effect is increased by increasing contrast in permeability, but the normal stress alteration due to elastic contrast increases rapidly as the rupture velocity approaches the generalized Rayleigh velocity. Because the alteration in effective normal stress due to either effect can be positive or negative, depending on the contrast in properties, the two effects can augment or offset each other. Citation: Rudnicki, J. W., and J. R. Rice (2006), Effective normal stress alteration due to pore pressure changes induced by dynamic slip propagation on a plane between dissimilar materials, J. Geophys. Res., 111, B10308, doi:10.1029/2006JB004396. 1. Introduction extended side is more permeable. An increase in pore pressure reduces the effective compressive stress and hence [2] A number of recent field studies [Chester et al., 1993; the frictional resistance to slip, whereas a decrease has the Chester and Chester, 1998; Lockner et al., 2000; Wibberley opposite effect. Rudnicki and Koutsibelas [1991] considered and Shimamoto, 2003; Sulem et al., 2004; Noda and the case of identical properties on the two sides of a Shimamoto, 2005] have identified the ultracataclastic cores completely impermeable fault plane. Following a sugges- of mature fault zones, where the slip is concentrated, and tion of J. R. Rice (personal communication, 1987), they shown that the core is much less permeable to flow across it argued that the pore pressure increase, rather than the than is the adjoining material of the damage zone. Because decrease on the other side of the slip zone, affects the rupture slip causes compression on one side of the fault wall and propagation. Their interpretation emerges as the proper limit extension on the other, the pore pressure tends to increase case here when there is a much more permeable material on on the compressive side and decrease on the extensile. This the compressive side than on the extensile. strong gradient results in a pore pressure on the fault slip [3] This paper calculates the pore pressure induced by a surface (treated as a plane) that depends on the difference in dynamically propagating fault within the framework of the properties on the two sides. The pore pressure on the fault model used by Rice et al. [2005] (hereinafter referred to as increases substantially when the compressed side is signif- RSP). The calculation is based on a model of discontinuous icantly more permeable, and decreases when, instead, the slip on a plane in an otherwise homogeneous poroelastic solid, but, in calculating the pore pressure change, we 1 Department of Mechanical Engineering and Department of Civil and include the effects of differences in material properties in Environmental Engineering, Northwestern University, Evanston, Illinois, narrow damage and granulation zones along the fault walls. USA. 2 Department of Earth and Planetary Sciences and Division of The pore pressure discontinuity predicted to occur across a Engineering and Applied Sciences, Harvard University, Cambridge, completely impermeable slip plane idealizes the spatially Massachusetts, USA. rapid pore pressure variation that would occur across a narrow but finite width fault zone with, generally, different Copyright 2006 by the American Geophysical Union. properties than the surrounding material (Figure 1). The 0148-0227/06/2006JB004396$09.00 B10308 1 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 Figure 1. Schematic diagram showing that the pore pressure change is equal in magnitude and opposite in sign on the two sides of the slip plane. Inset shows a more elaborate near-fault model in which the fault plane is the boundary between two materials that may be different from the material farther from the fault. The layers shown in the inset are so narrow that they are idealized as experiencing a uniform fault-parallel strain exx with the same magnitude but opposite signs on the two sides of the boundary (where slip occurs). model treated in the main part of the paper considers the and opposite in sign on the two sides of the material material on the two sides of the fault as homogeneous and boundary. These are the same strains as would be present simply assumes, as in the work by Rudnicki and Koutsibelas if the homogeneous material outside extended all the way to [1991], that the pore pressure on the compressive side the slip plane. When the material on the compressive side of controls strength. We show, however, in Appendix B that the fault is much more permeable than that on the extensile a more elaborate model of the near fault region can be side, the model reduces to the impermeable slip plane included simply by modifying the coefficient of the pore idealization with pore pressure on the compressive side pressure (Skempton’s coefficient) in the main text. controlling strength. [4] In the more elaborate model (Figure 1, inset), the [5] A result of the analysis is that the alteration of pore permeability and poroelastic properties of the material on pressure and hence of effective normal stress on the fault is the two sides of the fault differ from each other and from the proportional to the along fault gradient of the slip. This is homogeneous material farther from the fault. This model is the same form as the alteration of normal stress due to consistent with fault zone studies that show that the fault heterogeneous slip between dissimilar elastic solids, an core is embedded in a damaged region that may extend effect that has been widely studied in seismology [Weertman, several meters beyond the core [Chester et al., 1993; 1980; Andrews and Ben-Zion, 1997; Harris and Day, 1997; Chester and Chester, 1998; Wibberley and Shimamoto, Cochard and Rice, 2000; Ben-Zion, 2001; Xia et al., 2005]. 2003] and that the slip surface is often coincident with the Consequently, we are able to compare the magnitude of the boundary between ultracataclasites of different origin (from two effects. Their combination provides a more general host rocks on the two sides) or at a boundary between one framework for the inclusion of material heterogeneities, not of the ultracataclasites and damaged host rock [Chester and only dissimilarity of the crustal blocks on the two sides of Chester, 1998]. These near-fault regions are, however, the fault but also the dissimilarity of permeability and idealized as sufficiently narrow that they are subjected to poroelastic properties on the two sides of the slip surface. fault parallel strains that are uniform but equal in magnitude Because both effects may be of either sign, decreasing or 2 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 Figure 2. Schematic illustration of the cohesive zone, slip pulse model used by RSP. increasing the effective normal stress on the fault, depend- but they suggest ratios fd/fs in the range 0.2 to 0.8 with a ing on the direction of slip and mismatch of properties, they preference for the lower end of the range. Slip continues at can augment or offset each other. constant stress t r until the trailing edge of the slip zone at x = L. No further slip accumulates for x L and stress 2. Formulation increases above t r. RSP assume a linear decrease of shear stress from t p to t r with distance behind the leading edge of [6] RSP considered a propagating slip pulse of length L the slip zone. If L ! 1, the model reduces to the semi- with slip weakening as shown in Figure 2. The pulse is infinite slip zone considered by Poliakov et al. [2002]. As propagating at a constant velocity v. The stress on the slip discussed by RSP, when the slip zone is much larger than plane far ahead of the slipping zone is s0xy. The frictional the end zone size, L R, as in the Poliakov et al. [2002] resistance to slip on the fault plane is given by model, the initial stress s0xy is not significantly different 8 from the residual friction stress t r . < t p þ t p t r ð x=RÞ; R x < 0 [7] Because the fault is fluid saturated, and we are t ¼ t 0 ð xÞ ¼ ð1Þ interested in changes relative to the ambient state of stress : tr ; L x < R s0ij and pore pressure p0 before rupture, we will write s yy(x, 0) in (2) as syy(x, 0) + Dp where Dp is the change in pore The resistance weakens from a peak strength t p at initiation pressure from the ambient value; this reinterprets syy(x, 0) of slip to a residual strength t r at which large slip can occur. as the ambient effective normal stress s0yy + p0; the reinter- In a common interpretation (we mention a different one preted term syy(x, 0), written syy for shortness subsequently, below), these strengths are assumed to be entirely frictional is constant during the rupture. Our goal is to calculate the in origin such that t can be expressed as effect of the pressure induced by dynamic propagation on the frictional resistance. In order to make the problem t ¼ f s yy ð x; 0Þ ¼ f syy ð x; 0Þ þ pð x; 0Þ ð2Þ simply tractable, we assume that the weakening from t p to t r is unaffected by changes in effective normal stress, as where f is a friction coefficient, syy(x, 0) is the total normal if it represented a true shear cohesion that was weakened by stress on the slip plane and s yy(x, 0) is the effective normal slip and ultimately lost. The part of the strength normally stress (both positive in tension), and p(x, 0) is pore pressure. denoted by t r is wholly frictional in origin and is directly In the case considered in RSP, both p(x, 0) and syy(x, 0) proportional to effective normal stress. Hence this part of were constant, and hence also s yy(x, 0); in the generalization the strength is variable along the fault. Thus we assume of this paper, only syy(x, 0) is constant, and that only when we consider materials of identical elastic properties on the t ¼ t 0 ð xÞ fr Dp ð3Þ two sides of the fault. Immediately in front of the slipping zone (at the onset of slip), the resistive stress is t = t p, where fr is a constant and now t r = frsyy(x, 0) in (1). where t p = fss yy(x, 0) and fs is a static coefficient of Effectively, then, we are considering a strength relation in friction. The frictional resistance decreases with increasing the form slip and at a distance R behind the edge of the slipping zone reaches a residual value given by t r = fds yy(x, 0), where fd t ¼ cðd Þ fr syy þ p ð4Þ is a dynamic coefficient of friction. A typical value of the static coefficient of friction for rock is fs = 0.6. As discussed where c(d) is the cohesive part of strength, assumed to by RSP, there is less certainty about appropriate values of fd, weaken with slip d. For simplicity, the particular function 3 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 c = c(d) is chosen to make c(d) vary linearly with x, from only alteration of the effective normal stress is the pore c(0) at the rupture tip x = 0 to 0 at x = R. In that pressure induced by the near slip plane dissimilarity of interpretation, t p t r is simply to be regarded as a way of properties, then it is reasonable to assume, as did Rudnicki writing c(0). and Koutsibelas [1991], that the pore pressure increase [8] During dynamic slip propagation, there is insufficient controls propagation. However, this effect is likely to act time for pore fluid diffusion and conditions are undrained in combination with others, such as that due to elastic except for the small but critical boundary layer effect along dissimilarity of material farther from the fault discussed in the fault walls that is addressed in Appendix B. This effect the next paragraph. Consequently, we will present results takes place over a spatial scale perpendicular to the fault that for near slip zone arrangement of properties that both is typically on the order of a few millimeters to a few tens of increase and decrease pore pressure. millimeters. When the slip surface is idealized as a com- [10] If the materials away from the near fault region are pletely impermeable plane, undrained conditions pertain modeled as elastic, but with different properties on the two right up to the surface itself and the change in pore pressure sides of the fault, then inhomogeneous slip at the interface is related to total stress changes Dsij by induces a change in normal stress Comninou [1978] and Adams [1995, 1998] in addition to the change in effective 1 normal stress caused by the pore pressure (i.e., Dsyy 6¼ 0 on Dp ¼ B Dsxx þ Dsyy þ Dszz ð5Þ 3 y = 0, as assumed above). This change may also be positive or negative depending on the direction of slip, the direction where B is Skempton’s coefficient. In an infinite, isotropic of propagation and the mismatch of properties. In a later and homogeneous linear elastic solid, slip on a straight, section, we compare the alteration of normal stress in this planar fault induces no change in the normal stress on the case to the alteration of effective normal stress due to pore fault plane and, consequently, Dsyy vanishes on y = 0. For pressure and show that the net change in effective normal plane strain, the change of the out-of-plane normal stress is stress is the combination of both effects. given there by Dszz = n uDsxx where n u is the undrained [11] We also note that neglecting pore pressure effects but Poisson’s ratio. Thus the change in pore pressure is assuming a linear dependence of the friction coefficient, or simply of the shear strength, on the slip rate d_ leads to a 1 linear relation of the same form as (7) between the changes Dpð x; 0þ Þ ¼ Bð1 þ n u ÞDsxx ð x; 0þ Þ ð6Þ 3 in shear stress Dsxy(x, 0+) and the fault parallel stress change Dsxx(x, 0+) on the fault plane. This follows from where the superscript plus indicates evaluation on y = 0 as it the connection between the slip rate and the fault parallel is approached through positive values. For right-lateral slip, normal strain for steady propagation (see (B1) and the the pore pressure will increase on the positive side of the x following parenthetical remark). Although we do not treat axis. Substituting (6) into (3) and subtracting the initial, this case, the solution could be obtained from that here. In ambient shear stress s0xy yield the change in shear stress on _ particular, for a friction coefficient of the form fr = fr0 + fr1d, the plus side of the slip plane as equations in the same form as equations (7) and (8) hold, but now with t r = fr0syy and with frB(1 + n)/3 in Dsxy ð x; 0þ Þ ¼ 1 fr Bð1 þ n u ÞDsxx ð x; 0þ Þ þ gð xÞ ð7Þ equation (7) replaced by fr1v(1 n)s0yy/m. (An analogous 3 solution with velocity strengthening friction has been developed recently by Brener et al. [2005] for self-healing where slip pulses at a nonopening interface between deformable and rigid solids.) gð xÞ ¼ t 0 ð xÞ s0xy 8 > s0 t r þ t p t r ð1 þ x=RÞ; < R x < 0 3. Solution xy ¼ > : [12] RSP have shown that the change in total stresses for s0xy t r ; L x < R steady propagation of a Mode II rupture can be written as ð8Þ follows in terms of a single analytic function M(z) of the complex variable z [9] In Appendix B, we show that the more elaborate Dsxx ¼ Dsxx Dp model of the near fault material leads to an expression for the change in pore pressure that is identical to (6) but with B ¼ 2as Im 1 a2s þ 2a2d M ðzd Þ 1 þ a2s M ðzs Þ =D replaced by B0 = BW/w, where B is now to be interpreted as ð9aÞ the Skempton coefficient outside the near fault border regions (as n u is the undrained Poisson’s ratio of this region) and w and W (given in Appendix B) depend on the poroelastic properties and permeabilities of the near fault syy Dp ¼ 2as 1 þ a2s Im½ M ðzd Þ M ðzs Þ =D Dsyy ¼ D materials to either side (inset of Figure 1). Thus, in all the ð9bÞ following expressions, B is replaced by B0. Because W may be either positive or negative, the pore pressure may increase, promoting slip by decreasing the effective com- i 2 pressive normal stress, or decrease, inhibiting slip by sxy ¼ Re½4as ad M ðzd Þ 1 þ a2s M ðzs Þ =D Dsxy ¼ D ð9cÞ increasing the effective compressive normal stress. If the 4 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 the real and imaginary parts of M+(x) in (12) can be written as 2Re½M þ ð xÞ ¼ M þ ð xÞ þ M ð xÞ ð14aÞ 2iIm½M þ ð xÞ ¼ M þ ð xÞ M ð xÞ ð14bÞ Substituting these into (12) and rearranging give ð1 þ kiÞM þ ð xÞ þ ð1 kiÞM ð xÞ ¼ 2gð xÞ on L x 0 ð15Þ The problem has been reduced to finding the function M(z) that is analytic everywhere in the cut plane and approaches values on either side of the cut L x 0 that are related by (15) with (8). This is a type of problem that arises commonly in complex variable formulations of elasticity. Figure 3. Velocity-dependent portion of k (first bracket in Detailed discussions are given by Muskhelishvili [1992] and (13)) against the rupture velocity v divided by the shear England [2003]. The solution is obtained by a modified wave speed cs for two values of the undrained Poisson’s version of the procedure used by RSP and is described in ratio, 0.25 and 0.40. Appendix A. Results for a mathematically similar problem qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arising in supershear rupture propagation are given in the where z = x + qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d iady, z s = x + ias y, ad = 1 ðv=cd Þ2 , as = auxiliary material of Dunham and Archuleta [2005]. [15] The desired function is 1 ðv=cs Þ2 , cd and cs are the dilatational and shear wave speeds, and Re and Im stand for the real and imaginary cosðpeÞ 12e 1 parts. The denominator is the Rayleigh function M ð zÞ ¼ s0xy t r t p t r z ð z þ LÞ2þe p Z 0 2 ð1 þ t=RÞdt D ¼ 4as ad 1 þ a2s ð10Þ 1e 1 ð16Þ R ðt Þ2 ðt þ LÞ2þe ðt zÞ which vanishes when v equals the Rayleigh wave speed. (RSP used the notation stot where e is given by ij for what we call the total stress sij here, and used the notation sij for what we call the 1 effective stress s ij here.) e¼ arctanðk Þ ð17Þ [13] Evaluating (9a) and (9c) on the slip plane y = 0 yields p 4as a2d a2s When e = 0 (which occurs for B = B0 = 0) the expression for Dsxx x; 0 ¼ Im M ð xÞ ð11aÞ M(z) reduces to that of RSP. Because D = 0 (see (13)) when D the rupture speed v equals the Rayleigh wave velocity (0.92 cs when n u = 0.25 and 0.94 cs for n u = 0.4), in this Dsxy x; 0 ¼ Re M ð xÞ ð11bÞ limit k becomes unbounded and e = ±1/2, depending on the sign of B0. (Recall that for B0 < 0 the extended side of the where the superscript plus or minus indicates the limit as fault is more permeable than the compressed side.) At v = 0, y = 0 is approached through positive or negative values. the first bracket in (13) equals 2. Thus both k and e have Substituting into (7) then gives finite values at zero velocity and the effects of the pore pressure persist at low velocities, if not so low that the description of the field as being undrained almost every- Re½M þ ð xÞ ¼ kIm½M þ ð xÞ þ gð xÞ on L x 0 ð12Þ where loses validity. Figure 3 plots the velocity-dependent portion of k (first bracket in (13)) against the rupture speed v where, using equation (6) with B0 replacing B, as discussed divided by the shear wave speed cs for two values of the following equation (8) undrained Poisson’s ratio n u = 0.25 and 0.40. The second bracket vanishes if the fluid is very compressible (B = B0 = 0) 2 0 and is equal to one half in the limit of incompressible solid 4as ad a2s B ð1 þ n u Þ k ¼ fr ð13Þ and fluid constituents for a homogeneous material (B = B0 = 1 D 3 and n u = 1/2). Thus, for the latter limit and zero velocity k is simply equal to fr . [16] Figure 4 shows the variation of e with v/cs for three [14] Because the shear stress is continuous on the entire values of the friction coefficient (0.6, 0.4, and 0.2) and two plane y = 0 (including the slipping region), the values of the effective Skempton coefficient B0 (0.9 and 0.5) same arguments used by RSP can be used to show that (z), where M (z) is defined by M (z) = M ðzÞ and the for n u = 0.4. If B0 < 0, the values of e are the negative of M(z) = M those shown. Since the decrease of the friction coefficient overbar denotes the complex conjugate. As a consequence, from a static to a dynamic value is neglected in the portion 5 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 Figure 4. Magnitudes of e equal to 0.4 and 0.45 correspond to rupture speeds ranging from 0.90cs (for B0 = 0.9, fr = 0.6) to 0.94cs (nearly the Rayleigh wave speed cr = 0.94cs for n u = 0.4) for the parameters used in Figure 4. [19] For positive values of e, the pore pressure increases rapidly at the onset of slip at x = 0 and the largest pore pressure increase is induced in the end zone near the front of the slipping zone. The magnitude of the induced pressure increases with e and frDp(x, 0+) can exceed 80% of Figure 4. Variation of e with v/cs for undrained Poisson’s ratio n u = 0.4, three values of the friction coefficient and two values of the effective Skempton coefficient B0. multiplying the pore pressure (3) or is regarded as a separate cohesive term (4), the appropriate value of fr should lie between the static and dynamic values and, likely, closer to the dynamic (residual) value. The higher value of B0 might be more appropriate for a highly comminuted and disag- gregated fault zone that is much more permeable on the compressive side. However, as discussed in the Appendix B, a smaller contrast in permeability and a lower shear modulus for the near fault material on the compressive side (relative to the modulus for the less damaged material farther away) tends to reduce the effective value of Skemp- ton’s coefficient. This reduction is reflected by the choice of B0 = 0.5. Figure 3 shows that the velocity-dependent portion of k (and e) does not depend strongly on n u and otherwise n u enters only as a product with B0 in the sum 1 + n u. 4. Pore Pressure [17] Combining (6) and (11a) and using the expression for k (13) give the change of pore pressure on the positive side of the y axis fr Dpð x; 0þ Þ ¼ kImfM þ ð xÞg ð18Þ where Im{M+(x)} is the imaginary part of M(z) as y approaches zero through positive values in L x 0. This can be calculated numerically directly from (16) but is also given by (A8) of the Appendix A. Pore pressure changes for positive and negative values of e with magnitudes equal to 0.1, 0.2, 0.3, 0.4, and 0.45 are plotted for L/R = 2.0 and L/R = 5.0 in Figure 5. [18] For the six cases plotted in Figure 4, the magnitude of e at zero velocity ranges from a few per cent (0.03 for B0 = 0.5 and fr = 0.2) to 0.15 for B0 = 0.9 and fr = 0.6. For B0 = 0.9, e = 0.2 corresponds to v/cs equal to 0.59, 0.76 and Figure 5. Induced pore pressure Dp(x, 0+) multiplied by 0.87 for fr = 0.6, 0.4 and 0.2, respectively; for B0 = 0.5, the friction coefficient fr and divided by the cohesive zone e = 0.2 corresponds to v/cs equal to 0.80, 0.86 and 0.891 for stress drop c(0) = t p t r. Results are shown for (a) L/R = fr = 0.6, 0.4 and 0.2, respectively. A magnitude of e = 0.3 2.0 and (b) for L/R = 5.0. Curves are labeled by positive and corresponds to rupture speeds of 0.81cs, 0.87cs, 0.91cs, negative values of e for five magnitudes: 0.1, 0.2, 0.3, 0.4, 0.88cs, 0.916cs and 0.93cs for the six cases plotted in and 0.45. 6 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 [22] For e < 0, the magnitude of the pore pressure decreases slowly from its largest value near the end of the weakening zone (x = R) until very close to the trailing edge of the slip zone where it drops abruptly to zero. (The different behaviors of the pore pressure at the leading and trailing edges for positive and negative e are evident from the effects on the exponents of x and L + x in (A8)). Thus the pore pressure decrease increases the effective value of the frictional resistance by a substantial fraction of t p t r for the larger magnitudes of e. For negative e, the magnitude of the pore pressure change is greater over a larger proportion of the slipping zone. Figure 6 shows that the effect is more dramatic for larger L/R. [23] The induced pore pressure alters the effective shear resistance, t 0(x) frDp(x, 0+), and changes its distribution on the slip zone. The effective shear resistance minus the residual resistance t r (divided by t p t r) is shown for the Figure 6. Dependence of the pore pressure changes on same values of e in Figure 7 for L/R = 2.0 and 5.0. For L/R. Results are shown for e = ±0.3 and L/R = 1.1, 2.0, comparison the nominal, linear distribution of shear resis- 5.0 and the limit L/R ! 1. tance in the absence of pore pressure change, t 0(x), minus t r, is also plotted. As shown, for e > 0, the induced pore pressure increase causes a much more precipitous drop in c(0) = t p t r for e = 0.45. Since e increases with the shear resistance with distance back from the edge of the increasing velocity, the induced pore pressure contributes slipping zone (x = 0). As already noted, the pore pressure to velocity weakening. The distribution of pore pressure is causes a reduction in the effective residual shear resistance more sharply peaked in the end zone for the larger values of on the slipping region outside the end zone (negative values e. For negative values of e, the pore pressure decreases in Figure 7 for L x < R). For e < 0, the pore pressure roughly linearly with the onset of slip at x = 0 and achieves decrease causes a more gradual decrease in the shear its largest decrease near the end of the slip weakening zone resistance until very near the trailing edge of the slip zone (x = R). As for positive e, the magnitude of the change and causes the shear resistance to remain closer to t p over increases with the magnitude of e and frDp(x, 0+) is about the entire slipping region (rather than dropping to t r). For 95% of t p t r for e = 0.45. Because increasing magnitude both positive and negative e, the effect is larger for larger of e corresponds to increasing velocity, the increasing magnitudes and hence larger velocities. For negative e, the magnitude of the pore pressure decrease (increasing effective effective residual shear resistance is increased much more compressive stress) inhibits rupture propagation. than it is decreased for positive e. [20] The maximum induced pore pressure depends weakly on L/R and is about the same in Figures 5a and 5b. This is 5. Energy Release Rate shown more clearly in Figure 6, which plots the induced pore pressure frDp (divided by c(0) = t p t r) for e = ±0.3 [24] In this section, we calculate and discuss the effect of and L/R = 1.1, 2.0, 5.0 and the limit L/R ! 1. The the induced pore pressure on the energy required to drive magnitude of the pore pressure change induced in the end the fault at given a velocity (for a given nominal slip zone increases only slightly with L/R. weakening relation). For steady state propagation of the [ 21 ] The magnitudes of the pore pressure changes slip zone, there can be no change in the strain energy or induced outside the end zone (L x < R) differ kinetic energy. Consequently, as noted by RSP, the work of significantly for positive and negative values of e. For e > 0, the applied stress on the total relative slip must equal the a pore pressure increase with a magnitude roughly 10 to energy dissipated against the frictional resistance 20% of t p t r is induced on the slipping zone outside the Z end zone. The magnitude is slightly larger and the decay dT s0xy dT ¼ t 0 ðdÞ fr Dp dd ð19Þ toward the end of the slip zone (x = L) is slower for the 0 smaller values of e. Figure 6 shows that the magnitude of the pore pressure induced outside the end zone, although a where d T = d(x = L) is the total relative displacement small fraction of t p t r, increases with increasing L/R. accumulated at the trailing edge of the slipping zone and t 0 The pore pressure increase induced on L < x < R reduces is regarded as a function of slip accumulated behind the tip the effective value of the residual friction stress. For a d rather than position x (see section 6). Using Dsyy = 0, maximum magnitude of frDp outside the end zone about subtracting t rd T from both sides, and noting that t 0 t r = 0 20% of t p t r and the range of t r /t p = 0.2 – 0.8 for d(x = R) d d T gives (corresponding to the range of the ratio of static to dynamic values of the friction coefficient fs/fd cited earlier from Z d ð x¼RÞ Z dT RSP), frDp ranges from 0.05 to 0.80 times t r. Thus the s0xy t r dT ¼ t 0 ðdÞ t r dd fr Dpdd ð20Þ 0 0 effective frictional resistance remains positive although it can be reduced to as little as 20% of its nominal value. 7 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 where Z 1 R ð1 pÞ S ;e ¼ 1 þe dp ð23Þ L 1 0 p2e ½1 ð R=LÞp 2 When e = 0, the integral can be solved exactly to give the result in RSP and in the limit R/L ! 0, S(0; e) = 1/[(1/2 + e) (3/2 + e)]. Figure 8 plots (22) with (23) for various values of e. For a fixed value of R/L and positive e, the induced pore pressure increase reduces the driving stress s0xy t r for a given cohesive zone stress drop; for negative e the induced reduction of pore pressure increases the driving stress. [26] In the Appendix A, we also show that the total locked-in displacement can be expressed as 1 2 t p t r cosðpeÞ R 2þe R dT ¼ L D ;e ð24Þ mF ðvÞ L L where F(v) = D(v)/as(1 a2s ) as in RSP and Z 1 R ð1 pÞ½ð1=2 þ eÞ ð R=LÞp D ;e ¼ 1 dp ð25Þ L p2e ½1 ð R=LÞp 2þe 1 0 In the limit R/L ! 0, D(0; e) = 1/(1/2 + e). When e = 0, the integral can be solved exactly to give the result in RSP and when v approaches the Rayleigh wave speed so that e = 1/2, D(R/L; 1/2) = 1/2. Substituting for the stress ratio from (22) and for d T from (24) gives the following expression for Gnom: 2 1þ2e 2 t p t r cosðpeÞ R Gnom ¼ L Dð R=L; eÞSð R=L; eÞ p mF ðvÞ L ð26Þ Gnom reduces to RSP (their equation (18)) for e = 0 and in the limit R/L ! 0, Gnom/(md 2T/pL) reduces to F(v)/(1 + e) which also agrees with RSP for e = 0. Figure 7. Effective frictional resistance minus t r (divided by c(0) = t p t r) for (a) L/R = 2.0 and (b) L/R = 5.0. Curves are labeled by positive and negative values of e. Also shown is the nominal slip weakening in the absence of pore pressure changes t 0. [25] The left side of (20) is the nominal energy supplied by the applied loads in excess of the work against the residual friction stress t r: s0xy t r Gnom ¼ tp tr dT ð21Þ tp tr where we have divided and multiplied by (t p t r). In Appendix A, we show that the scaled stress drop ratio can be written as 1 Figure 8. Stress drop scaled by the cohesive stress drop s0xy t r cosðpeÞ R 2þe R ¼ S ;e ð22Þ c(0) = t p t r against R/L for various values of e (values to tp tr p L L the right of the curves). 8 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 (in the absence of pore pressure), the effect of an increase in pore pressure is to reduce the energy that must be supplied to drive the fault. Conversely, a decrease in pore pressure increases the energy that must be supplied. [28] Using dd = (@d/@x)dx and changing the limits of integration in the second term on the right in (20) yield Z L @d Gnom ¼ GB¼0 fr Dpð x; 0þ Þ ð xÞdx ð29Þ 0 @x Substituting for Dp(x, 0+) from (18), for @d/@x from (A13) and then from (A15) gives 2 2kL 1 Gnom ¼ GB¼0 t p t r cosðpeÞ mF ðvÞ p Z 1 L R 2 H x ; dx ð30Þ 0 R L where we have used the change of variable x = x/L in the integral. Thus the second term on the right side gives the amount that the nominal energy needed to drive the fault for a fixed material fracture energy (GB=0) is reduced by an induced pore pressure increase (e, k > 0) or increased by a pore pressure decrease (e, k < 0). Dividing by GB=0 and using (26) give the ratio Gnom 1 ¼ ð31Þ GB¼0 1 þ X where R 1 L R 2 k 0 H R;L dx Figure 9. Nominal energy supplied by the applied loads X ¼ ð32Þ Gnom divided by the material value in the absence of porous pð R=LÞ1þ2e Dð R=L; eÞSð R=L; eÞ media effects GB=0 against R/L for (a) five positive values of the velocity-dependent parameter e = 0.1, 0.2, 0.3, 0.4 This ratio is unity in the absence of pore fluid effects (B = 0) and 0.45 and (b) four negative values 0.1, 0.2, 0.3, and thus for positive e gives the fraction by which the and 0.4. energy that must be supplied to drive the fault is reduced; for negative e the ratio exceeds unity and gives the proportion by which the energy must be increased. [27] In the absence of induced pore pressure or when the [29] Figure 9 plots (31) as a function of R/L for positive Skempton coefficient, B, is zero, Gnom is equal to the (Figure 9a) and negative (Figure 9b) values of e. Thus, for energy dissipated against friction in excess of t r in the end e > 0, the energy that must be supplied to drive the fault zone, R x 0, which is the first term on the right in decreases with e and increasing propagation velocity. (20): Recall that for the values of B, and fr used in Figure 4, the magnitude of e at zero velocity ranges from a few per Z d ð x¼RÞ cent to 0.15. A magnitude of e = 0.3 corresponds to 0 rupture speeds of 0.81cs, 0.87cs, 0.91cs, 0.88cs, 0.916cs GB¼0 ¼ t ðd Þ t r dd ð27Þ 0 and 0.93cs for the six cases plotted in Figure 4. Magni- tudes of e equal to 0.4 and 0.45 correspond to rupture Within the idealization here that t 0 is unaffected by the pore speeds ranging from 0.90cs to 0.94cs (nearly the Rayleigh pressure, we regard GB=0 as a material parameter. For the wave speed cr = 0.94cs for n u = 0.4) for the parameters alternative interpretation of the slip weakening as purely used in Figure 4. Consequently, the reduction is substan- cohesive, expressed by (4), tial, and is more than 50%, for rupture speeds in excess of about 0.5cs. Z d ð x¼RÞ [30] There is an increase in (31) with R/L, which is greater GB¼0 ¼ cðdÞdd ð28Þ for smaller values of e. Since the peak induced pore 0 pressure depends only weakly on R/L, this increase reflects the more rapid decline in pore pressure outside the end zone with c(d) = 0 for d d(x = R). Thus, for a given value of for the larger values of R/L (see Figure 6). GB=0, reflecting a given t versus d relation in the end zone 9 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 because the nominal energy release rate (for B = 0) is interpreted as a material parameter. As in RSP, the dependence on L/R is very weak and plots for the other values of L/R used (1.1, 1.5, 2.0., and 10.0) are virtually indistinguishable from Figure 10. 7. More General Perspective on Material Dissimilarity Effects in Dynamic Rupture [34] In this section we compare the alteration of effective normal stress, due to induced pore pressure on the fault plane, with the change in normal stress induced by spatially inhomogeneous, mode II sliding on a plane between elastic solids with different material properties [Comninou, 1978; Adams, 1995, 1998]. The latter effect has been studied extensively in seismology [Weertman, 1980; Andrews and Ben-Zion, 1997; Harris and Day, 1997; Cochard and Rice, 2000; Ben-Zion, 2001; Xia et al., 2005]. In particular, we Figure 10. Relation for the cohesive part of the stress drop show how differences in poroelastic properties in thin layers c(d), divided by c(0) = t p t r, versus relative slip, along the fault (as in Figure 1 and in Appendix B) modify multiplied by c(0)/GB=0, implied by the solution. Shown for the interpretation of material dissimilarity as it has been L/R = 5.0 and several values of e. considered thus far. [35] We adopt the formulation of Weertman [1980] for steadily traveling slip distributions of form d = d(x vt) on the interface between homogeneous elastic half-spaces to [31] For negative values of e, the reduction in pore illustrate the effect. In that formulation, the shear and pressure dramatically increases the energy needed to drive normal stresses are the fault, by ratios exceeding 2, except when the slip weakening zone is a large fraction of the total slip zone Z þ1 ðvÞ m ddðx0 Þ=dx0 0 length (R L). For fixed R/L, the energy required sxy ð xÞ ¼ s0xy dx ð33Þ increases with velocity and hence would tend to inhibit p 1 x x0 propagation. The very large increases in energy required for small R/L reflect the large decreases in pore pressure induced outside the end zone (L x R) as shown in syy ð xÞ ¼ s0yy m* ðvÞddð xÞ=dx ð34Þ Figures 5 and 6. The functions, labeled m (v) and m*(v) by Weertman, are 6. Implied Slip Weakening Law defined in terms of his additional functions of rupture speed v [32] The frictional shear stress t 0(x) in the absence of pore pressure has been assumed to decrease linearly with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distance behind the rupture edge as shown schematically in ai ¼ 1 v2 =2c2si ; bi ¼ 1 v2 =c2si ; g i ¼ 1 v2 =c2di ð35Þ Figure 2. Because the resulting relative displacements on the slip zone can be calculated, as noted by RSP (following Palmer and Rice [1973]), the distribution of t 0(x) implies a with i = 1 or 2. Weertman’s subscript 1 refers to the relation between t 0 and the slip d. In the absence of induced material in y > 0 (where we previously denoted the near- pore pressure, this relation is independent of rupture speed. fault material by plus), and 2 refers to that in y < 0 (with Although the relation is not linear and depends on R/L, RSP near-fault material denoted minus above). The a here show (by comparing results for the limiting cases of R/L = 0 should not be confused with earlier uses of that symbol; b and R/L = 1) that the departure from linearity is small and and g correspond to as and ad, respectively, as introduced that the dependence on R/L is weak for a fixed material earlier. In terms of those functions [Weertman, 1980], with fracture energy (their G and corresponding to our GB=0). misprint corrections by Cochard and Rice [2000], m (v) and [33] In contrast to RSP, the slip weakening relation m*(v) are here depends on the rupture velocity (and porous media parameters) through e. Figure 10 shows the effective 2m1 m2 ¼ m m g 1 a22 g 1 b1 a41 shear resistance against the relative slip (multiplied by D1 þ D2 1 2 c(0)/GB=0) for L/R = 5.0 and e = ±0.2 and ±0.45. The þm2 g 1 1 a21 g 2 b2 a42 ð36Þ relation is also plotted for e = 0, corresponding to B = 0 and the case considered by RSP. As shown the decrease (increase) in pore pressure for e < (>)0 causes the curve 2m1 m2 to drop more (less) rapidly for small displacements. The m* ¼ m1 g 1 b1 a41 g 2 b 2 a22 D1 þ D2 areas under the curves are required to be identical (equal m2 g 2 b2 a42 g 1 b1 a21 ð37Þ to one for the normalization used) by (27) and (28) 10 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 where to (33) and (34) shows that the extensional strains along the fault walls are D1 ¼ m1 m2 1 a21 1 a22 g 2 b1 þ g 1 b1 a21 g 2 b2 a22 1 þ c dd 1 c dd þ m22 ð1 g 1 b1 Þ g 2 b2 a42 ð38Þ þ xx ¼ xx;1 ¼ ; xx ¼ xx;2 ¼ ð41Þ 2 dx 2 dx where D2 ¼ m1 m2 1 a21 1 a22 g 1 b2 þ g 1 b1 a21 g 2 b2 a22 þ m21 ð1 g 2 b2 Þ g 1 b1 a41 ð39Þ c ¼ ðD1 D2 Þ=ðD1 þ D2 Þ ð42Þ c reverses sign when we exchange one half-space for the In the references cited, D1 + D2 above is written simply as other or, for a given position of the two materials, reverse D, but both D1 and D2 are needed for the present purposes. the rupture propagation direction. In terms of those [36] When the two materials are identical, as in the expressions, the result in the Appendix B for the pore treatment of the materials far from the fault in the earlier pressure change on the fault plane, (B9) with (B10), is now part of this paper, m? = 0 and altered to mðgb a4 Þ W ðvÞ dd ðvÞ ¼ m ð40Þ pf ¼ ð43Þ bð1 a2 Þ 2 dx where (dropping the subscript i). When v ! 0, this m = m(1 c2s /c2d) = m/[2(1 n)] where, in the present undrained context, n Z þ wþ Z w Z þ wþ þ Z w corresponds to n u. The combination g ib i a4i is the W ðvÞ ¼ þ þ cðvÞ Z þZ Zþ þ Z Rayleigh function for material i; that is, it vanishes (other Z w =m þ Z w =m þ þ þ than at v = 0) at v = cRi. Thus (40) shows that when the two m* ðvÞ ð44Þ half-spaces are identical, m (v) = 0 at their common Rayleigh Zþ þ Z speed cR. [37] When the half-spaces are dissimilar, a generalized The first term in W(v) is the same as before and independent Rayleigh speed cGR is defined as the value of v > 0, if such of velocity. The second term, proportional to c(v), results exists, for which m vanishes, i.e., m (cGR) = 0. Such a cGR because the along fault extensional strains are no longer of exists for modest dissimilarity of properties, typically for identical magnitude on the two sides. The third term, shear wave speeds different by less than 20 – 30%, a proportional to m?(v), occurs because a nonzero change in condition that often seems to be met for natural faults syy is induced in the bimaterial case. The coefficients of [Andrews and Ben-Zion, 1997]. When v = cGR, (33) shows c(v) and m?(v) are averages of w and w/m, on the two that nonuniform slip does not alter the shear stress but sides of the fault, weighted by Z±, respectively. Thus the (34) shows that it does alter the normal stress, in a tensile equations for pore pressure and effective stress, correspond- direction when m* > 0. (Note that d_ = vdd/dx, so that if ing to (33) and (34), are now d_ 0 and v > 0, as implicitly assumed here, dd/dx 0.) Exchanging the two half-spaces for one another or, equiv- W ðvÞ dd ð xÞ pð xÞ ¼ p0 ; ð45Þ alently, running the same slip history in the opposite 2 dx direction along the interface, changes the sign of m*. In general, m* > 0 if the wave speeds of material 1 are less than those of material 2, and vice versa; expressed differently 0 0 ? W ðvÞ dd ð xÞ syy ð xÞ þ pð xÞ ¼ syy þ p m ðvÞ þ ; ð46Þ [Andrews and Ben-Zion, 1997], normal stress clamping is 2 dx decreased by nonuniform slip when the direction of prop- agation of the slip pattern is the same as the direction of the The latter shows that the proper measure of the propensity fault wall shear displacement in the slower material. for slip to alter effective normal stress is the sum of the [38] Weertman [1980] suggested the possibility of, and Weertman m? and the poroelastic W/2 derived here. Either Adams [1998] and Rice [1997] made explicit, simple term may be positive or negative depending on the solutions of (33) and (34), together with a Coulomb friction dissimilarity of properties adjacent to or farther from the law syx = fsyy with constant f, for which pulses of slip slip zone, the direction of propagation and the sense of propagate at cGR. They involve constant d_ in all sliding slip. regions, when 0 < s0yx < fs0yy [see Cochard and Rice, [40] As an example, consider material 1 (or plus) to be 2000]. slightly more compliant and to have a slightly lower shear [39] The dissimilarity of elastic material away from the wave velocity than material 2 (orpminus). ffiffiffi In particular, cs1 = fault on the two sides, in addition to differences very near 0.90cs2 and cd1/cs1 = cd2/cs2 = 3, corresponding to m1 = the fault predicted in the inset of Figure 1, alters the 0.75m2, r1 = 0.923r2 and n 1 = n 2 = 0.25. For this choice of calculation of pore pressure described in Appendix B. In Poisson’s ratio (here to be interpreted as the undrained particular, the condition that the fault parallel strains are of value), the Rayleigh wave speed in each material is 0.92 equal magnitude, +xx = xx, is no longer satisfied. Instead, a times the respective shear wave velocity. The generalized further analysis of the Weertman [1980] derivations leading Rayleigh wave velocity, at which m (cGR) = 0, is cGR = 11 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 of k+b +/kb > 1, so that the first (constant) term in W is >0, correspond to a more permeable compressive side of the fault; values less than one, making that first W term less than zero, correspond to a more permeable extensile side. (In general, that is, for different values of the shear modulus and Skempton’s coefficients on the two sides of the fault, the sign of the first term in (44) for W depends on the full mismatch of properties, not just on the ratio k+b +/kb ). Even if the materials bounding either side of the fault have identical properties, there is still a pore pressure induced if there is elastic mismatch farther from the fault (i.e., the contribution to the pore pressure then comes entirely from the latter two terms in (44)). [42] Figure 12 shows that the alteration of the effective normal stress due to the induced pore pressure change may be of either sign and is comparable in magnitude to the alteration due to the elastic mismatch. The rapid increase of m?/m2 as v approaches cGR suggests, however, that the effect Figure 11. Plot of c, equation (42), against v/cs2 up to v = cGR of the elastic mismatch will dominate near this limit, at least for m1 = 0.75m2, r1 = 0.923r2 and pffiffiffin 1 = n 2 = 0.25 so that if B is not too large (larger values of B increase W) and cs1 = 0.90cs2, cd1/cs1 = cd2/cs2 = 3 and cGR = 0.87cs2. mismatch in permeability is not extreme. If the half-spaces are exchanged so that the material in y > 0 is less compliant (has a greater shear wave speed), then the same plot (Figure 12) 0.87cs2 for these choices. Figure 11 plots c (42), and applies with the sign of the vertical axis reversed and the Figure 12 plots m?(v)/m2 as functions of velocity (divided values of k+b+/kb replaced by their reciprocals. Thus, in by cs2) up to cGR. In this case both c and m? are positive, but general, any of the terms due entering (44) may be positive exchanging the two materials or reversing the direction of or negative and the sign of the term due to near fault slip introduces a negative sign. The magnitudes of both c heterogeneity (first in (44)) may differ from that of terms and m increase as v approaches cGR, but both are finite there due to elastic dissimilarity (sum of last two entering (44)). (c = 0.155, m?/m2 = 0.178). Therefore the sign and magnitude of the alteration of the [41] Figure 12 also plots values for W/2m2 in order to effective normal stress depends not only on the elastic compare the magnitudes of the bimaterial and porous media dissimilarity of the materials bounding the fault but also on effects on alteration of the normal stress. As discussed in their differences from the material very near the slip surface Appendix B, the properties entering W pertain to the near- and the differences of the poroelastic properties of this fault material. Consequently, we choose the shear moduli, m+and m, entering W, (B10) with (B3), to be the same and equal to the average of the two shear moduli away from the fault, called m here, and the Poisson’s ratios again to be 0.25. Although B is often taken to be 0.9 for fault gouge [Roeloffs and Rudnicki, 1985; Rudnicki, 2001], we note in Appendix B that differences in properties on the two sides of the fault and the likelihood of greater damage near the fault tends to reduce the magnitude of the effective value of B although it may be of either sign. In addition, Roeloffs [1988] has discussed evidence for decreases in B with increasing effective stress, suggesting that smaller values are more appropriate for earthquake depths. Consequently, we have taken B = 0.6 for the material on both sides of the fault in plotting Figure 12. For this example, because the near properties are assumed to differ only in the values of the product kb, (44) simplifies to þ Z Z * ðvÞ=m W ðvÞ ¼ w þ cðv Þ m ð47Þ Figure 12. Plot of m*(v)/m2 against v/cs2 for the same elastic Zþ þ Z mismatch as in Figure 11: m1 = 0.75m2, r1 = 0.923r2 and pffiffiffi where w+ = w = w and m+ = m = m. Results are shown in n 1 = n 2 = 0.25 so that cs1 = 0.90cs2, cd1/cs1 = cd2/cs2 = 3 Figure 12 for values of the ratio k+b +/kb equal to 1, 10, and cGR = 0.87cs2. Remaining curves show W/2m2 for different 5, 2 and 1, 0.5, 0.2, 0.1 and 0. Since m?(cGR)/m = 0.204 values of the ratio of the product of the near fault permeability exceeds c(cGR) = 0.155, the sum of the latter two terms and compressibility on the two sides of the fault, k+b+/kb . changes from positive to negative and causes the slight On both sides of the fault, the near fault shear modulus is taken downturn in values of W(v)/2 as v approaches cGR. Values to be the average of values farther from the fault, B = 0.6 and n = 0.25. 12 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 material on the two sides of the zone of concentrated from differences in the properties of this near fault material sliding. from that farther from the fault. Although parameter values are uncertain, a comparison of the magnitude of the two 8. Discussion effects suggests that the effect due to nonuniform slip between different elastic solids may dominate as the rupture [43] We have calculated the pore pressure induced by slip velocity approaches the generalized Rayleigh speed for the propagation for a model that idealizes recent detailed studies bimaterial, at least if the effective Skempton’s coefficient is of fault zone structure [Chester et al., 1993; Chester and not too large, and the ratio of permeabilities is neither very Chester, 1998; Lockner et al., 2000; Wibberley and Shimamoto, large nor very small. 2003; Sulem et al., 2004; Noda and Shimamoto, 2005]. In [46] Even when the effect due to elastic dissimilarity of particular, slip is modeled as occurring on a plane bounded material away from the fault is dominant, pore pressure by material with permeabilities and poroelastic properties changes due to near fault heterogeneity may augment or that are different on each side of the slip plane and from the counteract the bimaterial effect. The sign of both effects can properties of the elastic material farther from the fault. The be positive or negative depending on the heterogeneity and pore pressure change is the result of poroelastic deformation the direction of slip and propagation. For example, the of the fault wall; changes due to inelastic porosity changes, normal compressive stress is reduced by nonuniform slip dilation or compression, of the fault zone material would be when the direction of propagation of the slip distribution is in addition to these. Although the detailed calculations are the same as the direction of the fault wall displacement in carried out for the limiting case in which the compressive the slower material. This reduction might, however, be side of the slip zone is much more permeable than the diminished by a increase in effective compressive stress, extensile side, we have shown that they can be applied to due to a decrease in pore pressure, if the extensile side of the more elaborate model simply by modifying the effective the fault is more permeable. Thus the net effect of slip on value of the Skempton coefficient. In particular, by chang- the effective normal stress will depend on the details of the ing the sign of the effective Skempton’s coefficient from properties, both elastic and poroelastic, of the material positive to negative treats the case in which the extensile bounding the zone of concentrated slip. side of the fault is more permeable. [47] Here the slip zone is assumed to occur at the [44] Induced pore pressure by poroelastic compression interface of the two materials. Fault zone studies [Chester discussed here is one of a number of mechanisms that have et al., 1993; Chester and Chester, 1998; Lockner et al., been suggested for dynamic weakening of slip resistance 2000; Wibberley and Shimamoto, 2003; Sulem et al., 2004; during earthquakes. These include thermal pressurization of Noda and Shimamoto, 2005] do show that the principal slip pore fluid [Lachenbruch, 1980; Mase and Smith, 1987; surface is very narrow, less than about 1 to 5 mm, and thus Garagash and Rudnicki, 2003a, 2003b; Garagash et al., reasonably idealized as a planar discontinuity for some 2005; Rice, 2006], flash heating of asperity contacts [Rice, purposes. Such studies also show that the relatively imper- 1999; Tullis and Goldsby, 2003; Rice, 2006] and others meable fault core is a wider zone of 10 mm to hundreds of [Sibson, 1975; Spray, 1993, 1995; Goldsby and Tullis, millimeters and bounded by a more permeable, damaged 2002; Chambon et al., 2002]. For the most part, these zone (grading to undamaged material farther from the fault). require rapid slip to generate heat sufficiently rapidly and The model shown in the inset of Figure 1 is consistent with relatively large slip to generate sufficiently high temperature slip occurring at the boundary of these two zones. If only (although Segall and Rice [2006] have shown that shear the effect of near fault heterogeneity is considered, and the heating can be significant toward the end of the nucleation slip zone is plausibly assumed to take a path of least period, before slip velocities become seismic). Although the resistance where the effective compressive normal stress is mechanism discussed here increases in magnitude with least, then the results here indicate that the slip zone will increasing velocity of propagation, it is also operative at choose a path where the compressive side is more perme- small slip and at low velocities. Consequently, it may be a able. For the direction of slip (right lateral) and propagation factor in allowing sufficient slip to occur long enough for (to the right) shown in Figure 1, this is consistent with the other mechanisms to come into play or for preventing positive material being the less permeable ultracataclastic incipient slip from progressing, depending on the sign of core and the negative material being the adjacent more the induced pore pressure change. permeable, damage layer. Of course, this simple picture [45] The model provides a more general framework for could be complicated by a variety of other effects. Never- considering the effects of material heterogeneities perpen- theless, in a recent numerical study of the bimaterial effect, dicular to the fault on alterations of the effective normal Brietzke and Ben-Zion [2006] found that when several slip stress. Previous studies [Weertman, 1980; Andrews and surfaces were possible, the rupture tended to choose a Ben-Zion, 1997; Harris and Day, 1997; Cochard and Rice, material interface where the compressive normal stress 2000; Ben-Zion, 2001; Xia et al., 2005] have focused on the was reduced. It is possible to speculate that when the pore alteration of normal stress induced by inhomogeneous slip pressure changes due to material heterogeneity are included, at the interface between elastic solids with different prop- the rupture tends to choose the interface where the reduction erties. The calculations here show that alterations of com- of effective normal stress is greatest, though this is certainly parable magnitude in the effective normal stress can result an issue in need of further study. from pore pressure changes induced by heterogeneous [48] Another issue in need of further study is the effect of properties. More specifically, pore pressure changes result near fault damage induced by slip propagation. In contrast from differences in the permeability and poroelastic prop- to the scenario of the previous paragraph, such damage may erties of the material on the two sides of the slip zone and have an effect which, typically, acts oppositely to the 13 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 reduction of compressive normal stress due to far field is straightforward to verify that the complex function z/(z + elastic dissimilarity, by inducing negative pore pressure L)(1/2)e, with branch cut taken on y = 0, L x 0, changes (suctions) along the slip surface. If there is no or satisfies (15) with zero right side. only very small far field material dissimilarity, then the [51] Because any analytic function multiplied by this suctions will dominate and cause an increase in the effective function is also a solution to the homogeneous equation normal compression, and hence partial stabilization of the and because it will be convenient to have a solution that fault. This possibility arises because a variety of studies decays as z1 as jzj ! 1, we take the homogeneous [Poliakov et al., 2002; Kame et al., 2003; Rice et al., 2005; solution to be Andrews, 2005; Ben-Zion and Shi, 2005] have shown that unless the direction of maximum principal stress is at an 1 cð zÞ ¼ ðA1Þ unusually shallow angle with the fault, say, less than 25 ð z þ LÞ 1 2e 1 ð zÞ2þe degrees, then stresses predicted near the rupture front are expected to cause Mohr-Coulomb failure and damage Dividing both sides of (15) by c+(x) and using the preferentially on the extensile side of the slipping plane. homogeneous equation yields Such preference is supported by field evidence [Poliakov et al., 2002]. Ben-Zion and Shi [2005] have shown this same M ð xÞ þ M ð xÞ 2gð xÞ tendency for damage on the extensile side in simulations of ¼ ðA2Þ rupture along an interface between elastically dissimilar cð xÞ cð xÞ ð1 þ kiÞcþ ð xÞ materials. The spatial extent of the damaged zone is predicted to decrease with depth [Rice et al., 2005] but Equation (A2) has the solution [Muskhelishvili, 1992; the preferred side for damage remains. Presumably, this England, 2003] damage would increase the permeability on the extensile Z 0 side whereas, conversely, compression of the other side may M ð zÞ 1 g ðt Þ dt A reduce the permeability there. The results here indicate that ¼ þ þ ðA3Þ cð zÞ pið1 þ kiÞ L c ðt Þ t z p this effect, unless compensated by decrease of total com- pressive stress due to far field material dissimilarity, would decrease the pore pressure and hence increase the effective where A is a constant. compressive stress and the energy required to propagate the [52] The constant A is determined by the condition that fault. Ben-Zion and Shi [2005] have shown that including the singularity in M(z) vanish as z ! 0. This results from the the effects of damage induced by rupture can modify and physical requirement that the slip weakening zone causes partially stabilize some aspects of the bimaterial effect. the relative displacement to vanish smoothly at the edge of Further study of the effect on damage on altering the near the slipping zone. Setting z(1/2)+eM(z) equal to zero in the fault permeability structure is needed. limit z ! 0 yields Z 1 9. Conclusion 0 gðt Þð L þ t Þ2e A ¼ cosðpeÞ 1 dt ðA4Þ L ðt Þ 2e [49] Interaction of pore fluid with material heterogeneity near the slip zone that is representative of that observed in pffiffiffiffiffiffiffiffiffiffiffiffiffi fault zones can affect rupture propagation. An increase of where cos(pe) = 1/ 1 þ k 2 . The slip must, however, also cease pore pressure that reduces the effective compressive stress smoothly at the trailing edge of the slipping zone, x = L. and facilitates slip propagation occurs if the compressive This leads to the additional requirement on A that results side of the slip zone (modeled as a plane) is more permeable from setting (z + L)(1/2)eM(z) equal to zero in the limit than the tensile side; conversely, a decrease of pore pressure z ! L: that increases the effective compressive stress and inhibits Z 1 slip propagation occurs if tensile side is more permeable. 0 gðt Þðt Þ2þe A ¼ cosðpeÞ 1 dt ðA5Þ Although a more complex and realistic model will undoubt- L ðt þ LÞ2þe edly alter the details of the calculations here, the main conclusion that near fault heterogeneity affects rupture Equating the two expressions (A4) and (A5) for A yields the propagation is unlikely to change. Understanding of how following constraint on g(t) the particular effect studied here, pore pressure changes due to heterogeneous poroelastic properties, interacts with a Z 0 gðt Þ variety of other effects must await further modeling and 1 1 dt ¼ 0 ðA6Þ L 2e ðt Þ ðt þ LÞ2þe observational studies. Substituting from (A4) for A and c(z) into (A3) and then Appendix A: Details of the Solution using the constraint equation (A6) yield [50] The function M(z) is analytic everywhere in the cut Z plane and approaches values on either side of the cut L cosðpeÞ 12e 1 0 gðt Þdt x 0 that are related by (15). The solution proceeds by first M ð zÞ ¼ z ð z þ LÞ2þe 1e p 1 ðt Þ ðt þ LÞ2þe ðt zÞ L 2 finding a function that satisfies the homogeneous equation ((15) with zero right hand side). Although there is a formal ðA7Þ procedure for this [Muskhelishvili, 1992; England, 2003], it 14 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 When g(t) is substituted from (8), the integral in the term where multiplying (s0xy t r) can be done using contour integration by means similar to those described by RSP. 1 x R The resulting expression for M(z) is (16) in the text. The ImfM þ ð xÞg ¼ t p t r cosðpeÞH ; ðA15Þ p R L Im{M+(x)}, which is needed to obtain the pore pressure can be calculated numerically directly from (16) and is also [55] Evaluating d(x) from (A14) at x = L gives the total, given by locked-in displacement that has accumulated at the trailing edge of the slip zone dT. This total displacement can, ðcosðpeÞÞ2 1e 1 however, be obtained directly by the same arguments used ImfM þ ð xÞg ¼ t p t r ðxÞ ð L þ xÞ2þe 2 pR by RSP and Poliakov et al. [2002]. They show that d T is Z 1 ð1 sÞds proportional to A and is given by 1e 1 ðA8Þ 0 s ½ð L=RÞ s 2þe ½s þ ð x=RÞ 2 2Aas 1 a2s dT ¼ ðA16Þ where the change of variable t = sR has been used in the mD integral. Using (A4) for A and substituting the expression for g(t) A1. Stress Drop from (8) yield [53] Substitution of the expression for g into (A6) gives a relation between the ratio of the driving stress (s0xy t r) to Z 0 ð1 þ t=RÞð L þ t Þ2e 1 the cohesive zone stress drop (t p t r) and the scaled A ¼ t p t r cosðpeÞ 1 dt cohesive zone size R/L: R ðt Þ2e Z 1 0 ð L þ t Þ2e R0 h 1 1 i s0xy t r cosðpeÞ dt ðA17Þ 2e ðt þ LÞ2þe dt 1 s0xy t r R ð1 þ t=RÞ= ð t Þ L ðt Þ2e tp tr ¼ R0 h 1 1 i ðA9Þ 2e ðt þ LÞ2þe dt L 1=ðt Þ If the second integral is denoted I, then dI/dL is equal to (1/2 e) times the integral in (A10). Using this result and (22) The integral in the denominator can be evaluated by and combining the integrals make it possible to write A as converting it to a contour integral around the branch cut L x < 0 in the complex plane and the result is 12þe R R A¼L D ;e ðA18Þ Z 0 h i L L 1 1 1=ðt Þ2e ðt þ LÞ2þe dt ¼ p= cosðpeÞ ðA10Þ L where D(R/L; e) is given by (25). Substituting (A18) into (A16) gives (24) of the text. This expression for d T agrees Using (A10) and the substitution p = t/R in the integral in with (A14) evaluated at x = L. the numerator of (A9) gives (22) of the text. A2. Relative Slip Displacements Appendix B: Pore Pressure at the Fault Plane [54] The variation of the relative displacement can be [56] In the main text the fault has sometimes been written as described, for simplicity, as a completely impermeable plane of displacement discontinuity in a uniform material, @d ð1 n u Þ with pore pressure on the compressed side entering the ð xÞ ¼ 2xx ð x; 0þ Þ ¼ Dsxx ð x; 0þ Þ ðA11Þ @x m friction law. Although it is a reasonable approximation to assume uniform material properties at some distance away Substituting for Dsxx(x, 0+) from (9a) and using the identity from the fault, detailed examinations of fault zones [Chester (with Poisson’s ratio taken as the undrained value) et al., 1993; Chester and Chester, 1998; Wibberley and Shimamoto, 2003] show that the fault walls are often bor- 1 a2s dered by materials that are different from each other and 1 nu ¼ 2 ðA12Þ from the uniform material farther away. In this Appendix, we 2 ad a2s show the effect of these different near fault properties, give including the actual finite, if small, permeabilities of the materials involved (leading to a continuous pore pressure @d 2 variation across the fault plane), can be included by modifying ð xÞ ¼ ImfM þ ð xÞg ðA13Þ a parameter of the analysis based on discontinuous pressure @x mF ðvÞ at an impermeable plane in a uniform material. In particular, we assume that these regions of different properties border- where F(v) is defined following (24). Integrating and noting ing the fault are thin enough that they can be considered to that d = 0 at x = 0 give undergo a uniform x direction strain e+xx(t) on one side of the Z x fault and an equal and opposite strain e + xx(t) = exx(t) on the 2 tp tr s R other side (Figure 1, inset). Because those bordering zones dð xÞ ¼ cosðpeÞ H ; ds ðA14Þ pmF ðvÞ 0 R L are presumed to be thin (compared, say, to the along-strike 15 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 length scales R and L), the strains will be essentially identical [60] If the plane y = 0 is not completely impermeable, to those calculated along the fault walls for the uniform then spatially dependent pore pressure changes p(y, t) will material model in the body of the paper. These fault-parallel develop in each half-space. The difference between these strains are related to the slip rate d_ at a fixed position x and fields and the uniform, undrained pore pressure (B2) sat- rupture velocity v by isfies a homogeneous diffusion equation in each half-space [e.g., Rice and Cleary, 1976]: eþ _ xx ðt Þ ¼ exx ðt Þ ¼ d=2v ðB1Þ @ @ 2 pð y; t Þ p 0 ðt Þ ¼ ahy pð y; tÞ p 0 ðt Þ ðB5Þ where the compressive side of the fault is assumed to be @t @y 2 y > 0. (Because the solution is steady state 2e+xx = @d/@x = (1/v)@d/@t and this expression is consistent with (A11).) where ± again refer to y > 0 and y < 0, and a±hy are the The strains are calculated as in the main text, based on the hydraulic diffusivities. The hydraulic diffusivities can be solution for an impermeable fault plane in uniform material. written as k±/hfb ±, where k± are the permeabilities, b ± are [57] The hydraulic diffusivity ahy of fault gouge at the storage coefficients and hf is the fluid viscosity, which is representative seismogenic depths is generally estimated the same in both half-spaces. The diffusion equations must as being in the range of 1 – 10 mm2/s [Rice, 2006], and be solved in the two domains y > 0 and y < 0, subject to two the duration t of slip at a point in large earthquakes is conditions at the interface y = 0. The first is that the pore typically of the order of 1 s for every 1 m of slip [Heaton, fluid pressures be the same (i.e., p is continuous), 1990]. Thus the thickness of the region over which fluid diffusion smooths out the discontinuity of the impermeable pffiffiffiffiffiffiffiffi pð y ¼ 0þ ; t Þ ¼ pð y ¼ 0 ; t Þ ¼ pf ðt Þ ðB6Þ fault model, of the order of a few times ahy t , will not generally be larger than a few tens of millimeters. Thus we where pf (t) is the pore pressure on the fault that we seek. are concerned here with the properties and poromechanical The second is the fluid flux across the interface (given by response of border regions of that order of thickness along Darcy’s law) is continuous the fault walls. [58] Now consider a magnified view of such border @p @p regions along the fault plane, so that they appear as two kþ ð y ¼ 0þ ; t Þ ¼ k ð y ¼ 0 ; tÞ ðB7Þ @y @y half planes subjected to a uniform x direction strain e+xx(t) in y > 0 and an equal and opposite strain e + xx(t) = exx(t) in y < 0. The two half-spaces are assumed to be deforming under [61] Solution by Laplace transform shows that the pore plane strain conditions (ezz = 0) and are subjected to the pressure on the fault is given by same fault normal stress syy, which remains constant during slip. The shear stress sxy does vary but that does not affect Z þ pþ 0 ðt Þ þ Z p0 ðt Þ pf ðt Þ ¼ ðB8Þ the pore pressure under these conditions, assuming that the Z þ Z þ border regions are isotropic, or are aleotropic with principal pffiffiffiffiffiffiffiffiffiffiffi directions aligned with the x and y directions. What is the where Z± = k b . Substituting from (B2) and using + pore pressure pf (t) induced at the fault plane y = 0? exx(t) = exx(t) gives [59] To answer this question, first imagine that the plane y = 0 is completely impermeable. Then the pore pressure W dd pf ðt Þ ¼ W eþ xx ðt Þ ¼ ðB9Þ changes p±0(t) from the ambient value are uniform in each 2 dx half-space and given by where p 0 ðt Þ ¼ w exx ðt Þ ðB2Þ Z þ wþ Z w W¼ ðB10Þ The w± for each poroelastic half-space for undrained, plane Zþ þ Z strain conditions and constant syy is [Rice and Cleary, 1976] In the body of the paper we generalize the last pair of equations to the case of two elastically dissimilar half- B 1 þ n w ¼ 2m u ðB3Þ spaces adjoining the fault, each lined with a narrow fault- 3 1 n u bordering layer whose poroelastic properties define the w± and Z± as here. Substituting (B4) yields when the border regions are isotropic, where ± refer to the local, near fault properties in y b 0. The factor B±(1 + n ±u )/ 1 3(1 n ±u ) attains its maximum value, unity, for B± = 1 and pf ðtÞ ¼ ð BW =wÞð1 þ n u ÞDsþ xx ðt Þ ðB11Þ 3 n ±u = 1/2 which is the case when both solid and fluid constituents are incompressible. Note that the near fault where w here is defined as in (B3), B is the Skempton strains e±xx(t) (assumed spatially uniform in this magnified coefficient, and both are based on the properties farther view) are related to s±xx from the analysis of the text by the from the fault. Thus the fault pore pressure is defined by the uniform properties m and n u away from the fault, i.e., same relation as in a homogeneous material e xx ðt Þ ¼ ð1 n u Þs xx =2m ðB4Þ 1 pf ðt Þ ¼ B0 ð1 þ n u ÞDsþ xx ðt Þ ðB12Þ 3 16 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 where B0 = BW/w. Note, however, from (B10) that W need providing a check of some of the numerical calculations and also for helping uncover an error in our original analysis of the effects of elastic not be positive, and thus B0 may be of either sign, depending dissimilarity. J.W.R. is grateful for support from the Kavli Institute for on material properties in the two border regions. With this Theoretical Physics, Santa Barbara, while participating in the program on interpretation of B as B0, the solution for a uniform material Granular Physics and from the U.S. Department of Energy, Office of Basic with an impermeable fault plane, given in the main text, Energy Science, Geosciences Research Program. J.R.R. is grateful for support of NSF grants EAR-0125709 and 0510193 and of the NSF/USGS applies as well for the more realistic model of this Southern California Earthquake Center, funded by NSF Cooperative Appendix. The case B0 > 0 corresponds to weakening the Agreement EAR-0106924 and USGS Cooperative Agreement fault by induced pore pressure, and B0 < 0 to strengthening 02HQAG0008 (this is SCEC contribution 932). by induced pore suction. References [62] As a simple case, assume that all the near fault Adams, G. G. (1995), Self-excited oscillations of two elastic half-spaces properties, except for the permeabilities, are identical to sliding with a constant coefficient of friction, J. Appl. Mech., 62, 867 – those of the homogeneous material farther from the fault. 872. Then B0 reduces to Adams, G. G. (1998), Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip, J. Appl. Mech., 65, 470 – 475. ( pffiffiffiffiffiffiffiffiffiffiffiffiffi) Andrews, D. J. (2005), Rupture dynamics with energy loss outside the slip 0 1 k =k þ zone, J. Geophys. Res., 110, B01307, doi:10.1029/2004JB003191. B ¼B pffiffiffiffiffiffiffiffiffiffiffiffiffi ðB13Þ Andrews, D. J., and Y. Ben-Zion (1997), Wrinkle-like slip pulse on a fault 1 þ k =k þ between different materials, J. Geophys. Res., 102, 553 – 571. Ben-Zion, Y. (2001), Dynamic ruptures in recent models of earthquake faults, J. Mech. Phys. Solids, 49, 2209 – 2244. Hence, when the permeability of the extensional side is Ben-Zion, Y., and Z. Shi (2005), Dynamic rupture on a material interface much less than that of the compressional, k k+, then B0 = B. with spontaneous generation of plastic strain in the bulk, Earth Planet. If k/k+ = 102, which might be representative of having Sci. Lett., 236, 486 – 496. Brietzke, G., and Y. Ben-Zion (2006), Examining tendencies of in-plane ultracataclasite on the extensional side and a coarser gouge rupture to migrate to material interfaces, Geophys. Int. J., 167, or densely cracked material from the damaged fault core on doi:10.1111/j.1365-246X.2006.03137.x. the compressional side, based on properties inferred from Brener, E., S. Malinin, and V. Marchenko (2005), Fracture and friction: Lockner et al. [2000] and Wibberley and Shimamoto [2003], Stick-slip motion, Eur. Phys. J. E, 17, 101 – 113, doi:10.1140/epje/i2004- 10112-3. then B0 ’ 0.82B; if k/k+ = 101, B0 ’ 0.52B. So the effect Chambon, G., J. Schmittbuhl, and A. Corfdir (2002), Laboratory gouge of decreasing permeability of the part of the fault core on friction: Seismic-like slip weakening and secondary rate- and state- the extensional side is to reduce the effective value of effects, Geophys. Res. Lett., 29(10), 1366, doi:10.1029/2001GL014467. Chester, F. M., and J. S. Chester (1998), Ultracataclasite structure and Skempton’s coefficient in the solution. If the extensional friction processes of the Punchbowl Fault, San Andreas System, Califor- side is the more permeable, which is the case generally nia, Tectonophysics, 295(1 – 2), 199 – 221. expected based on where Mohr-Coulomb plasticity and Chester, F. M., J. P. Evans, and R. L. Biegel (1993), Internal structure and weakening mechanisms of the San Andreas fault, J. Geophys. Res., 98, damage is expected to occur most extensively near the 771 – 786. rupture front, then k > k+, and then B0 < 0 and suction Cochard, A., and J. R. Rice (2000), Fault rupture between dissimilar rather than pressure is induced on the fault plane. materials: Ill-posedness, regularization and slip-pulse response, J. Geophys. [63] More generally, if all properties of the border regions Res., 105, 25,891 – 25,907. Comninou, M. (1978), The interface crack in a shear field, J. Appl. Mech., differ from those farther away, so that (B13) does not apply, 45(2), 287 – 290. but if it is nevertheless the case that the extensional side is Dunham, E. M., and R. J. Archuleta (2005), Near-source ground motion essentially impermeable compared to the compressive, from steady state dynamic rupture pulses, Geophys. Res. Lett., 32, L03302, doi:10.1029/2004GL021793. k/k+ ! 0, then England, A. H. (2003), Complex Variable Methods in Elasticity, Dover, Mineola, N. Y. þ Garagash, D. I., and J. W. Rudnicki (2003a), Shear heating of a fluid- 0 mþþ 1 þ nþu = 1 nu B ¼B ðB14Þ saturated slip-weakening dilatant fault zone: 1. Limiting regimes, J. Geo- m ð1 þ n u Þ=ð1 n u Þ phys. Res., 108(B2), 2121, doi:10.1029/2001JB001653. Garagash, D. I., and J. W. Rudnicki (2003b), Shear heating of a fluid- saturated slip-weakening dilatant fault zone: 2. Quasi-drained regime, The same expression for B0 results, except that it is preceded J. Geophys. Res., 108(B10), 2472, doi:10.1029/2002JB002218. by a minus sign and all plus superscripts are changed to Garagash, D. I., D. G. Schaeffer, and J. W. Rudnicki (2005), Effect of rate minus, in the case for which the compressive side dependence in shear heating of a fluid-saturated fault zone, in Porome- is essentially impermeable compared to the extensional, chanics III-Biot Centennial (1905 – 2005), Proceedings, 3rd Biot Confer- ence on Poromechanics, edited by Y. Abousleiman, A. H.-D. Cheng, and k/k+ ! 1. The term in brackets involving Poisson’s ratio F.-J. Ulm, pp. 789 – 794, A. A. Balkema, Brookfield, Vt. varies by at most a factor of three. For n +u = 0.4 and n u = 0.2, Goldsby, D. L., and T. E. Tullis (2002), Low frictional strength of quartz this bracket is 1.55. Thus, in these cases with one side being rocks at subseismic slip rates, Geophys. Res. Lett., 29(17), 1844, doi:10.1029/2002GL015240. essentially impermeable compared to the other, B0 is roughly Harris, R., and S. M. Day (1997), Effects of a low velocity zone on a equal to the near fault Skempton’s coefficient B+ reduced by dynamic rupture, Bull. Seismol. Soc. Am., 87, 1267 – 1280. the ratio of the shear modulus near the fault to that farther Heaton, T. H. (1990), Evidence for and implications of self-healing pulses of slip in earthquake rupture, Phys. Earth Planet. Inter., 64, 1 – 20. away, or to B reduced by a similar factor. The effect is to Kame, N., J. R. Rice, and R. Dmowska (2003), Effects of prestress state and reduce the magnitude of the effective value of B, although rupture velocity on dynamic fault branching, J. Geophys. Res., 108(B5), the reduction might be partly offset by an increase in the 2265, doi:10.1029/2002JB002189. undrained Poisson’s ratio in the near fault material. Lachenbruch, A. H. (1980), Frictional heating, fluid pressure, and the resistance to fault motion, J. Geophys. Res., 85, 6097 – 6112. Lockner, D., H. Naka, H. Tanaka, H. Ito, and R. Ikeda (2000), Permeability [64] Acknowledgments. This work was begun during September and strength of core samples from the Nojima fault of the 1995 Kobe 2003 at the Isaac Newton Institute for Mathematical Sciences, University earthquake, in Proceedings of the International Workshop on the Nojima of Cambridge, England, while both authors were participants in the Fault Core and Borehole Data Analysis, Tsukuba, Japan, Nov 22 – 23, Program on Granular and Particle Laden Flows. We are grateful for the 1999, edited by H. Ito et al., U.S. Geol. Surv. Open File Rep., 00-129, support of the institute and the organizers. We thank Eric Dunham 147 – 152. 17 of 18 B10308 RUDNICKI AND RICE: PORE PRESSURE CHANGES DUE TO DYNAMIC SLIP B10308 Mase, C. W., and L. Smith (1987), Effects of frictional heating on the Rudnicki, J. W., and D. A. Koutsibelas (1991), Steady propagation of plane thermal, hydrologic, and mechanical response of a fault, J. Geophys. strain shear cracks on an impermeable plane in an elastic diffusive solid, Res., 92, 6249 – 6272. Int. J. Solids Struct., 27, 205 – 225. Muskhelishvili, N. I. (1992), Singular Integral Equations, 2nd ed., Dover, Segall, P., and J. R. Rice (2006), Does shear heating of pore fluid contribute Mineola, N. Y. to earthquake nucleation?, J. Geophys. Res., 111, B09316, doi:10.1029/ Noda, H., and T. Shimamoto (2005), Thermal pressurization and slip-weak- 2005JB004129. ening distance of a fault: An example of the Hanaore fault, southwest Sibson, R. H. (1975), Generation of pseudotachylyte by ancient seismic Japan, Bull. Seismol. Soc. Am., 95(4), 1224 – 1233. faulting, Geophys. J. R. Astron. Soc., 43, 775 – 794. Palmer, A. C., and J. R. Rice (1973), The growth of slip surfaces in the Spray, J. G. (1993), Viscosity determinations of some frictionally generated progressive failure of over-consolidated clay, Proc. R. Soc. London, Ser. silicate melts: Implications for fault zone rheology at high strain rates, A, 332, 527 – 548. J. Geophys. Res., 98, 8053 – 8068. Poliakov, A. N. B., R. Dmowska, and J. R. Rice (2002), Dynamic shear Spray, J. G. (1995), Pseudotachylyte controversy: Fact or friction?, Geo- rupture interactions with fault bends and off-axis secondary faulting, logy, 23, 1119 – 1122. J. Geophys. Res., 107(B11), 2295, doi:10.1029/2001JB000572. Sulem, J., I. Vardoulakis, H. Ouffroukh, M. Boulon, and J. Hans (2004), Rice, J. R. (1997), Slip pulse at low driving stress along a frictional fault Experimental characterization of the thermo-poro-mechanical properties between dissimilar media (abstract), Eos Trans. AGU, 78(46), Fall Meet. of the Aegion Fault gouge, C. R. Geosci., 336(4 – 5), 455 – 466. Suppl., F464. Tullis, T. E., and D. Goldsby (2003), Flash melting of crustal rocks at Rice, J. R. (1999), Flash heating at asperity contacts and rate-dependent almost seismic slip rates (abstract), Eos Trans. AGU, 84(46), Fall Meeting friction (abstract), Eos Trans. AGU, 80(46), Fall Meeting Suppl., F681. Suppl., Abstract S51B-05. Rice, J. R. (2006), Heating and weakening of faults during earthquake slip, Weertman, J. (1980), Unstable slippage across a fault that separates elastic J. Geophys. Res., 111, B05311, doi:10.1029/2005JB004006. media of different elastic constants, J. Geophys. Res., 85, 1455 – 1461. Rice, J. R., and M. P. Cleary (1976), Some basic stress diffusion solutions Wibberley, C. A. J., and T. Shimamoto (2003), Internal structure and per- for fluid-saturated elastic porous media with compressible constituents, meability of major strike-slip fault zones: The Median Tectonic Line in Rev. Geophys., 14, 227 – 241. Mie Prefecture, southwest Japan, J. Struct. Geol., 25, 59 – 78. Rice, J. R., C. G. Sammis, and R. Parsons (2005), Off-fault secondary Xia, K., A. J. Rosakis, H. Kanamori, and J. R. Rice (2005), Laboratory failure induced by a dynamic slip-pulse, Bull. Seismol. Soc. Am., 95(1), earthquakes along inhomogeneous faults: Directionality and supershear, 109 – 134, doi:10.1785/0120030166. Science, 308(5722), 681 – 684. Roeloffs, E. A. (1988), Fault stability changes induced beneath a reservoir with cyclic variations in water level, J. Geophys. Res., 93, 2107 – 2124. Roeloffs, E. A., and J. W. Rudnicki (1985), Coupled deformation diffusion J. R. Rice, Department of Earth and Planetary Sciences, Harvard effects on water-level changes due to propagating creep events, Pure University, 224 Pierce Hall, 29 Oxford Street, Cambridge, MA 02138, Appl. Geophys., 122, 560 – 582. USA. (rice@esag.deas.harvard.edu) Rudnicki, J. W. (2001), Coupled deformation-diffusion effects in the me- J. W. Rudnicki, Department of Civil and Environmental Engineering, chanics of faulting and failure of geomaterials, Appl. Mech. Rev., 54(6), Northwestern University, Evanston, IL 60208-3109, USA. (jwrudn@ 483 – 502. northwestern.edu) 18 of 18