Effective Normal Stress Alteration due to Pore Pressure Changes Induced by Dynamic Slip Propagation on a Plane Between Dissimilar Materials

[ 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.


Introduction
[2] A number of recent field studies [Chester et al., 1993;Chester and Chester, 1998;Lockner et al., 2000;Wibberley and Shimamoto, 2003;Sulem et al., 2004;Noda and Shimamoto, 2005] have identified the ultracataclastic cores of mature fault zones, where the slip is concentrated, and shown that the core is much less permeable to flow across it than is the adjoining material of the damage zone. Because slip causes compression on one side of the fault wall and extension on the other, the pore pressure tends to increase on the compressive side and decrease on the extensile. This strong gradient results in a pore pressure on the fault slip surface (treated as a plane) that depends on the difference in properties on the two sides. 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. An increase in pore pressure reduces the effective compressive stress and hence the frictional resistance to slip, whereas a decrease has the opposite effect. Rudnicki and Koutsibelas [1991] considered the case of identical properties on the two sides of a completely impermeable fault plane. Following a suggestion of J. R. Rice (personal communication, 1987), they argued that the pore pressure increase, rather than the decrease on the other side of the slip zone, affects the rupture propagation. Their interpretation emerges as the proper limit case here when there is a much more permeable material on the compressive side than on the extensile.
[3] This paper calculates the pore pressure induced by a dynamically propagating fault within the framework of the model used by Rice et al. [2005] (hereinafter referred to as RSP). The calculation is based on a model of discontinuous slip on a plane in an otherwise homogeneous poroelastic solid, but, in calculating the pore pressure change, we include the effects of differences in material properties in narrow damage and granulation zones along the fault walls. The pore pressure discontinuity predicted to occur across a completely impermeable slip plane idealizes the spatially rapid pore pressure variation that would occur across a narrow but finite width fault zone with, generally, different properties than the surrounding material ( Figure 1). The model treated in the main part of the paper considers the material on the two sides of the fault as homogeneous and simply assumes, as in the work by Rudnicki and Koutsibelas [1991], that the pore pressure on the compressive side controls strength. We show, however, in Appendix B that a more elaborate model of the near fault region can be included simply by modifying the coefficient of the pore pressure (Skempton's coefficient) in the main text.
[4] In the more elaborate model (Figure 1, inset), the permeability and poroelastic properties of the material on the two sides of the fault differ from each other and from the homogeneous material farther from the fault. This model is consistent with fault zone studies that show that the fault core is embedded in a damaged region that may extend several meters beyond the core [Chester et al., 1993;Chester and Chester, 1998;Wibberley and Shimamoto, 2003] and that the slip surface is often coincident with the boundary between ultracataclasites of different origin (from host rocks on the two sides) or at a boundary between one of the ultracataclasites and damaged host rock [Chester and Chester, 1998]. These near-fault regions are, however, idealized as sufficiently narrow that they are subjected to fault parallel strains that are uniform but equal in magnitude and opposite in sign on the two sides of the material boundary. These are the same strains as would be present if the homogeneous material outside extended all the way to the slip plane. When the material on the compressive side of the fault is much more permeable than that on the extensile side, the model reduces to the impermeable slip plane idealization with pore pressure on the compressive side controlling strength.
[5] A result of the analysis is that the alteration of pore pressure and hence of effective normal stress on the fault is proportional to the along fault gradient of the slip. This is the same form as the alteration of normal stress due to heterogeneous slip between dissimilar elastic solids, an effect that has been widely studied in seismology [Weertman, 1980;Andrews and Ben-Zion, 1997;Harris and Day, 1997;Cochard and Rice, 2000;Ben-Zion, 2001;Xia et al., 2005]. Consequently, we are able to compare the magnitude of the two effects. Their combination provides a more general framework for the inclusion of material heterogeneities, not only dissimilarity of the crustal blocks on the two sides of the fault but also the dissimilarity of permeability and poroelastic properties on the two sides of the slip surface. Because both effects may be of either sign, decreasing or 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 e xx with the same magnitude but opposite signs on the two sides of the boundary (where slip occurs).
increasing the effective normal stress on the fault, depending on the direction of slip and mismatch of properties, they can augment or offset each other.

Formulation
[6] RSP considered a propagating slip pulse of length L with slip weakening as shown in Figure 2. The pulse is propagating at a constant velocity v. The stress on the slip plane far ahead of the slipping zone is s xy 0 . The frictional resistance to slip on the fault plane is given by The resistance weakens from a peak strength t p at initiation of slip to a residual strength t r at which large slip can occur. In a common interpretation (we mention a different one below), these strengths are assumed to be entirely frictional in origin such that t can be expressed as where f is a friction coefficient, s yy (x, 0) is the total normal stress on the slip plane and s yy (x, 0) is the effective normal stress (both positive in tension), and p(x, 0) is pore pressure. In the case considered in RSP, both p(x, 0) and s yy (x, 0) were constant, and hence also s yy (x, 0); in the generalization of this paper, only s yy (x, 0) is constant, and that only when we consider materials of identical elastic properties on the 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 t p = Àf s s yy (x, 0) and f s is a static coefficient of friction. The frictional resistance decreases with increasing slip and at a distance R behind the edge of the slipping zone reaches a residual value given by t r = Àf d s yy (x, 0), where f d is a dynamic coefficient of friction. A typical value of the static coefficient of friction for rock is f s = 0.6. As discussed by RSP, there is less certainty about appropriate values of f d , but they suggest ratios f d /f s in the range 0.2 to 0.8 with a preference for the lower end of the range. Slip continues at constant stress t r until the trailing edge of the slip zone at x = ÀL. No further slip accumulates for x ÀL and stress 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 the slip zone. If L ! 1, the model reduces to the semiinfinite slip zone considered by Poliakov et al. [2002]. As discussed by RSP, when the slip zone is much larger than the end zone size, L ) R, as in the Poliakov et al. [2002] model, the initial stress s xy 0 is not significantly different from the residual friction stress t r .
[7] Because the fault is fluid saturated, and we are interested in changes relative to the ambient state of stress s ij 0 and pore pressure p 0 before rupture, we will write s yy (x, 0) in (2) as s yy (x, 0) + Dp where Dp is the change in pore pressure from the ambient value; this reinterprets s yy (x, 0) as the ambient effective normal stress s yy 0 + p 0 ; the reinterpreted term s yy (x, 0), written s yy for shortness subsequently, is constant during the rupture. Our goal is to calculate the effect of the pressure induced by dynamic propagation on the frictional resistance. In order to make the problem simply tractable, we assume that the weakening from t p to t r is unaffected by changes in effective normal stress, as if it represented a true shear cohesion that was weakened by slip and ultimately lost. The part of the strength normally denoted by t r is wholly frictional in origin and is directly proportional to effective normal stress. Hence this part of the strength is variable along the fault. Thus we assume where f r is a constant and now t r = Àf r s yy (x, 0) in (1). Effectively, then, we are considering a strength relation in the form where c(d) is the cohesive part of strength, assumed to weaken with slip d. For simplicity, the particular function c = c(d) is chosen to make c(d) vary linearly with x, from c(0) at the rupture tip x = 0 to 0 at x = ÀR. In that interpretation, t p À t r is simply to be regarded as a way of writing c(0).
[8] During dynamic slip propagation, there is insufficient time for pore fluid diffusion and conditions are undrained except for the small but critical boundary layer effect along the fault walls that is addressed in Appendix B. This effect takes place over a spatial scale perpendicular to the fault that is typically on the order of a few millimeters to a few tens of millimeters. When the slip surface is idealized as a completely impermeable plane, undrained conditions pertain right up to the surface itself and the change in pore pressure is related to total stress changes Ds ij by where B is Skempton's coefficient. In an infinite, isotropic and homogeneous linear elastic solid, slip on a straight, planar fault induces no change in the normal stress on the fault plane and, consequently, Ds yy vanishes on y = 0. For plane strain, the change of the out-of-plane normal stress is given there by Ds zz = n u Ds xx where n u is the undrained Poisson's ratio. Thus the change in pore pressure is where the superscript plus indicates evaluation on y = 0 as it is approached through positive values. For right-lateral slip, the pore pressure will increase on the positive side of the x axis. Substituting (6) into (3) and subtracting the initial, ambient shear stress s xy 0 yield the change in shear stress on the plus side of the slip plane as [9] In Appendix B, we show that the more elaborate model of the near fault material leads to an expression for the change in pore pressure that is identical to (6) but with B replaced by B 0 = BW/w, where B is now to be interpreted as 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 materials to either side (inset of Figure 1). Thus, in all the following expressions, B is replaced by B 0 . Because W may be either positive or negative, the pore pressure may increase, promoting slip by decreasing the effective compressive normal stress, or decrease, inhibiting slip by increasing the effective compressive normal stress. If the only alteration of the effective normal stress is the pore pressure induced by the near slip plane dissimilarity of properties, then it is reasonable to assume, as did Rudnicki and Koutsibelas [1991], that the pore pressure increase controls propagation. However, this effect is likely to act in combination with others, such as that due to elastic dissimilarity of material farther from the fault discussed in the next paragraph. Consequently, we will present results for near slip zone arrangement of properties that both increase and decrease pore pressure.
[10] If the materials away from the near fault region are modeled as elastic, but with different properties on the two sides of the fault, then inhomogeneous slip at the interface induces a change in normal stress Comninou [1978] and Adams [1995Adams [ , 1998 in addition to the change in effective normal stress caused by the pore pressure (i.e., Ds yy 6 ¼ 0 on y = 0, as assumed above). This change may also be positive or negative depending on the direction of slip, the direction of propagation and the mismatch of properties. In a later section, we compare the alteration of normal stress in this case to the alteration of effective normal stress due to pore pressure and show that the net change in effective normal stress is the combination of both effects.
[11] We also note that neglecting pore pressure effects but assuming a linear dependence of the friction coefficient, or simply of the shear strength, on the slip rate _ d leads to a linear relation of the same form as (7) between the changes in shear stress Ds xy (x, 0 + ) and the fault parallel stress change Ds xx (x, 0 + ) on the fault plane. This follows from the connection between the slip rate and the fault parallel normal strain for steady propagation (see (B1) and the following parenthetical remark). Although we do not treat this case, the solution could be obtained from that here. In particular, for a friction coefficient of the form f r = f r 0 + f r 1 _ d, equations in the same form as equations (7) and (8) hold, but now with t r = Àf r 0 s yy and with f r B(1 + n)/3 in equation (7) replaced by f r 1 v(1 À n)s yy 0 /m. (An analogous solution with velocity strengthening friction has been developed recently by Brener et al. [2005] for self-healing slip pulses at a nonopening interface between deformable and rigid solids.)

Solution
[12] RSP have shown that the change in total stresses for steady propagation of a Mode II rupture can be written as follows in terms of a single analytic function M(z) of the complex variable z Ds yy ¼ D s yy À Dp ¼ À2a Ds where , c d and c s are the dilatational and shear wave speeds, and Re and Im stand for the real and imaginary parts. The denominator is the Rayleigh function which vanishes when v equals the Rayleigh wave speed. (RSP used the notation s ij tot for what we call the total stress s ij here, and used the notation s ij for what we call the effective stress s ij here.) [13] Evaluating (9a) and (9c) on the slip plane y = 0 yields Ds where the superscript plus or minus indicates the limit as y = 0 is approached through positive or negative values. Substituting into (7) then gives where, using equation (6) with B 0 replacing B, as discussed following equation (8) [14] Because the shear stress is continuous on the entire plane y = 0 (including the slipping region), the same arguments used by RSP can be used to show that M(z) = M (z), where M (z) is defined by M (z) = M z ð Þ and the overbar denotes the complex conjugate. As a consequence, the real and imaginary parts of M + (x) in (12) can be written as Substituting these into (12) and rearranging give 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 ÀL x 0 that are related by (15) with (8). This is a type of problem that arises commonly in complex variable formulations of elasticity. Detailed discussions are given by Muskhelishvili [1992] and England [2003]. The solution is obtained by a modified version of the procedure used by RSP and is described in Appendix A. Results for a mathematically similar problem arising in supershear rupture propagation are given in the auxiliary material of Dunham and Archuleta [2005].
[15] The desired function is where e is given by When e = 0 (which occurs for B = B 0 = 0) the expression for M(z) reduces to that of RSP. Because D = 0 (see (13)) when the rupture speed v equals the Rayleigh wave velocity (%0.92 c s when n u = 0.25 and 0.94 c s for n u = 0.4), in this limit k becomes unbounded and e = ±1/2, depending on the sign of B 0 . (Recall that for B 0 < 0 the extended side of the fault is more permeable than the compressed side.) At v = 0, the first bracket in (13) equals 2. Thus both k and e have 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 everywhere loses validity. Figure 3 plots the velocity-dependent portion of k (first bracket in (13)) against the rupture speed v divided by the shear wave speed c s for two values of the undrained Poisson's ratio n u = 0.25 and 0.40. The second bracket vanishes if the fluid is very compressible (B = B 0 = 0) and is equal to one half in the limit of incompressible solid and fluid constituents for a homogeneous material (B = B 0 = 1 and n u = 1/2). Thus, for the latter limit and zero velocity k is simply equal to f r .
[16] Figure 4 shows the variation of e with v/c s for three values of the friction coefficient (0.6, 0.4, and 0.2) and two values of the effective Skempton coefficient B 0 (0.9 and 0.5) for n u = 0.4. If B 0 < 0, the values of e are the negative of those shown. Since the decrease of the friction coefficient from a static to a dynamic value is neglected in the portion multiplying the pore pressure (3) or is regarded as a separate cohesive term (4), the appropriate value of f r should lie between the static and dynamic values and, likely, closer to the dynamic (residual) value. The higher value of B 0 might be more appropriate for a highly comminuted and disaggregated 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 Skempton's coefficient. This reduction is reflected by the choice of B 0 = 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 B 0 in the sum 1 + n u .

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 where Im{M + (x)} is the imaginary part of M(z) as y approaches zero through positive values in ÀL 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 B 0 = 0.5 and f r = 0.2) to 0.15 for B 0 = 0.9 and f r = 0.6. For B 0 = 0.9, e = 0.2 corresponds to v/c s equal to 0.59, 0.76 and 0.87 for f r = 0.6, 0.4 and 0.2, respectively; for B 0 = 0.5, e = 0.2 corresponds to v/c s equal to 0.80, 0.86 and 0.891 for f r = 0.6, 0.4 and 0.2, respectively. A magnitude of e = 0.3 corresponds to rupture speeds of 0.81c s , 0.87c s , 0.91c s , 0.88c s , 0.916c s and 0.93c s for the six cases plotted in Figure 4. Magnitudes of e equal to 0.4 and 0.45 correspond to rupture speeds ranging from 0.90c s (for B 0 = 0.9, f r = 0.6) to 0.94c s (nearly the Rayleigh wave speed c r = 0.94c s 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 f r Dp(x, 0 + ) can exceed 80% of  c(0) = t p À t r for e = 0.45. Since e increases with increasing velocity, the induced pore pressure contributes to velocity weakening. The distribution of pore pressure is more sharply peaked in the end zone for the larger values of e. For negative values of e, the pore pressure decreases roughly linearly with the onset of slip at x = 0 and achieves its largest decrease near the end of the slip weakening zone (x = ÀR). As for positive e, the magnitude of the change increases with the magnitude of e and f r Dp(x, 0 + ) is about 95% of t p À t r for e = À0.45. Because increasing magnitude of e corresponds to increasing velocity, the increasing magnitude of the pore pressure decrease (increasing effective compressive stress) inhibits rupture propagation.
[20] The maximum induced pore pressure depends weakly on L/R and is about the same in Figures 5a and 5b. This is shown more clearly in Figure 6, which plots the induced pore pressure f r Dp (divided by c(0) = t p À t r ) for e = ±0.3 and L/R = 1.1, 2.0, 5.0 and the limit L/R ! 1. The magnitude of the pore pressure change induced in the end zone increases only slightly with L/R.
[21] The magnitudes of the pore pressure changes induced outside the end zone (ÀL x < ÀR) differ significantly for positive and negative values of e. For e > 0, a pore pressure increase with a magnitude roughly 10 to 20% of t p À t r is induced on the slipping zone outside the end zone. The magnitude is slightly larger and the decay toward the end of the slip zone (x = ÀL) is slower for the smaller values of e. Figure 6 shows that the magnitude of the pore pressure induced outside the end zone, although a small fraction of t p À t r , increases with increasing L/R. The pore pressure increase induced on ÀL < x < ÀR reduces the effective value of the residual friction stress. For a maximum magnitude of f r Dp outside the end zone about 20% of t p À t r and the range of t r /t p = 0.2 -0.8 (corresponding to the range of the ratio of static to dynamic values of the friction coefficient f s /f d cited earlier from RSP), f r Dp ranges from 0.05 to 0.80 times t r . Thus the effective frictional resistance remains positive although it can be reduced to as little as 20% of its nominal value.
[22] For e < 0, the magnitude of the pore pressure decreases slowly from its largest value near the end of the weakening zone (x = ÀR) 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 Àx 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 À 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) À f r Dp(x, 0 + ), and changes its distribution on the slip zone. The effective shear resistance minus the residual resistance t r (divided by t p À t r ) is shown for the same values of e in Figure 7 for L/R = 2.0 and 5.0. For comparison the nominal, linear distribution of shear resistance 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 the shear resistance with distance back from the edge of the slipping zone (x = 0). As already noted, the pore pressure causes a reduction in the effective residual shear resistance on the slipping region outside the end zone (negative values in Figure 7 for ÀL x < ÀR). For e < 0, the pore pressure decrease causes a more gradual decrease in the shear resistance until very near the trailing edge of the slip zone and causes the shear resistance to remain closer to t p over the entire slipping region (rather than dropping to t r ). For both positive and negative e, the effect is larger for larger magnitudes and hence larger velocities. For negative e, the effective residual shear resistance is increased much more than it is decreased for positive e.

Energy Release Rate
[24] In this section, we calculate and discuss the effect of the induced pore pressure on the energy required to drive the fault at given a velocity (for a given nominal slip weakening relation). For steady state propagation of the slip zone, there can be no change in the strain energy or kinetic energy. Consequently, as noted by RSP, the work of the applied stress on the total relative slip must equal the energy dissipated against the frictional resistance where d T = d(x = ÀL) is the total relative displacement accumulated at the trailing edge of the slipping zone and t 0 is regarded as a function of slip accumulated behind the tip d rather than position x (see section 6). Using Ds yy = 0, subtracting t r d T from both sides, and noting that t 0 À t r = 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 : where we have divided and multiplied by (t p À t r ). In Appendix A, we show that the scaled stress drop ratio can be written as 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 s xy 0 À 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 where F(v) = D(v)/a s (1 À a s 2 ) as in RSP and 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 G nom : G nom reduces to RSP (their equation (18)) for e = 0 and in the limit R/L ! 0, G nom /(md T 2 /pL) reduces to F(v)/(1 + e) which also agrees with RSP for e = 0.  [27] In the absence of induced pore pressure or when the Skempton coefficient, B, is zero, G nom is equal to the energy dissipated against friction in excess of t r in the end zone, ÀR x 0, which is the first term on the right in (20): Within the idealization here that t 0 is unaffected by the pore pressure, we regard G B=0 as a material parameter. For the alternative interpretation of the slip weakening as purely cohesive, expressed by (4), with c(d) = 0 for d ! d(x = ÀR). Thus, for a given value of G B=0 , reflecting a given t versus d relation in the end zone (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 Substituting for Dp(x, 0 + ) from (18), for @d/@x from (A13) and then from (A15) gives where we have used the change of variable x = Àx/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 (G B=0 ) is reduced by an induced pore pressure increase (e, k > 0) or increased by a pore pressure decrease (e, k < 0). Dividing by G B=0 and using (26) give the ratio This ratio is unity in the absence of pore fluid effects (B = 0) and thus for positive e gives the fraction by which the 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.
[29] Figure 9 plots (31) as a function of R/L for positive ( Figure 9a) and negative (Figure 9b) values of e. Thus, for e > 0, the energy that must be supplied to drive the fault decreases with e and increasing propagation velocity. Recall that for the values of B, and f r used in Figure 4, the magnitude of e at zero velocity ranges from a few per cent to 0.15. A magnitude of e = 0.3 corresponds to rupture speeds of 0.81c s , 0.87c s , 0.91c s , 0.88c s , 0.916c s and 0.93c s for the six cases plotted in Figure 4. Magnitudes of e equal to 0.4 and 0.45 correspond to rupture speeds ranging from 0.90c s to 0.94c s (nearly the Rayleigh wave speed c r = 0.94c s for n u = 0.4) for the parameters used in Figure 4. Consequently, the reduction is substantial, and is more than 50%, for rupture speeds in excess of about 0.5c s .
[30] There is an increase in (31) with R/L, which is greater for smaller values of e. Since the peak induced pore pressure depends only weakly on R/L, this increase reflects the more rapid decline in pore pressure outside the end zone for the larger values of R/L (see Figure 6). [31] For negative values of e, the reduction in pore pressure dramatically increases the energy needed to drive the fault, by ratios exceeding 2, except when the slip weakening zone is a large fraction of the total slip zone length (R % L). For fixed R/L, the energy required increases with velocity and hence would tend to inhibit propagation. The very large increases in energy required for small R/L reflect the large decreases in pore pressure induced outside the end zone (ÀL x ÀR) as shown in Figures 5 and 6.

Implied Slip Weakening Law
[32] The frictional shear stress t 0 (x) in the absence of pore pressure has been assumed to decrease linearly with distance behind the rupture edge as shown schematically in 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 relation between t 0 and the slip d. In the absence of induced pore pressure, this relation is independent of rupture speed. Although the relation is not linear and depends on R/L, RSP show (by comparing results for the limiting cases of R/L = 0 and R/L = 1) that the departure from linearity is small and that the dependence on R/L is weak for a fixed material fracture energy (their G and corresponding to our G B=0 ).
[33] In contrast to RSP, the slip weakening relation here depends on the rupture velocity (and porous media parameters) through e. Figure 10 shows the effective shear resistance against the relative slip (multiplied by c(0)/G B=0 ) for L/R = 5.0 and e = ±0.2 and ±0.45. The 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 to drop more (less) rapidly for small displacements. The areas under the curves are required to be identical (equal to one for the normalization used) by (27) and (28) 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.

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, 1995Adams, , 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 show how differences in poroelastic properties in thin layers along the fault (as in Figure 1 and in Appendix B) modify the interpretation of material dissimilarity as it has been considered thus far.
[35] We adopt the formulation of Weertman [1980] for steadily traveling slip distributions of form d = d(x À vt) on the interface between homogeneous elastic half-spaces to illustrate the effect. In that formulation, the shear and normal stresses are The functions, labeled m(v) and m*(v) by Weertman, are defined in terms of his additional functions of rupture speed v with i = 1 or 2. Weertman's subscript 1 refers to the material in y > 0 (where we previously denoted the nearfault material by plus), and 2 refers to that in y < 0 (with near-fault material denoted minus above). The a here should not be confused with earlier uses of that symbol; b and g correspond to a s and a d , respectively, as introduced earlier. In terms of those functions [Weertman, 1980], with misprint corrections by Cochard and Rice [2000], where In the references cited, D 1 + D 2 above is written simply as D, but both D 1 and D 2 are needed for the present purposes.
[36] When the two materials are identical, as in the treatment of the materials far from the fault in the earlier part of this paper, m ? = 0 and (dropping the subscript i). When v ! 0, this where, in the present undrained context, n corresponds to n u . The combination g i b i À a i 4 is the Rayleigh function for material i; that is, it vanishes (other than at v = 0) at v = c Ri . Thus (40) shows that when the two half-spaces are identical, m(v) = 0 at their common Rayleigh speed c R .
[37] When the half-spaces are dissimilar, a generalized Rayleigh speed c GR is defined as the value of v > 0, if such exists, for which m vanishes, i.e., m(c GR ) = 0. Such a c GR exists for modest dissimilarity of properties, typically for shear wave speeds different by less than 20 -30%, a condition that often seems to be met for natural faults [Andrews and Ben-Zion, 1997]. When v = c GR , (33) shows that nonuniform slip does not alter the shear stress but (34) shows that it does alter the normal stress, in a tensile direction when m* > 0. (Note that _ d = Àvdd/dx, so that if _ d ! 0 and v > 0, as implicitly assumed here, dd/dx 0.) Exchanging the two half-spaces for one another or, equivalently, running the same slip history in the opposite 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 [Andrews and Ben-Zion, 1997], normal stress clamping is decreased by nonuniform slip when the direction of propagation of the slip pattern is the same as the direction of the fault wall shear displacement in the slower material.
[38] Weertman [1980] suggested the possibility of, and Adams [1998] and Rice [1997] made explicit, simple solutions of (33) and (34), together with a Coulomb friction law s yx = Àfs yy with constant f, for which pulses of slip propagate at c GR . They involve constant _ d in all sliding regions, when 0 < s yx 0 < Àfs yy 0 [see Cochard and Rice, 2000].
[39] The dissimilarity of elastic material away from the fault on the two sides, in addition to differences very near the fault predicted in the inset of Figure 1, alters the calculation of pore pressure described in Appendix B. In particular, the condition that the fault parallel strains are of equal magnitude, xx + = À xx À , is no longer satisfied. Instead, a further analysis of the Weertman [1980] derivations leading to (33) and (34) shows that the extensional strains along the fault walls are c reverses sign when we exchange one half-space for the other or, for a given position of the two materials, reverse the rupture propagation direction. In terms of those expressions, the result in the Appendix B for the pore pressure change on the fault plane, (B9) with (B10), is now altered to The first term in W(v) is the same as before and independent of velocity. The second term, proportional to c(v), results because the along fault extensional strains are no longer of identical magnitude on the two sides. The third term, proportional to Àm ? (v), occurs because a nonzero change in s yy is induced in the bimaterial case. The coefficients of c(v) and Àm ? (v) are averages of w and w/m, on the two sides of the fault, weighted by Z ± , respectively. Thus the equations for pore pressure and effective stress, corresponding to (33) and (34), are now The latter shows that the proper measure of the propensity for slip to alter effective normal stress is the sum of the Weertman m ? and the poroelastic W/2 derived here. Either term may be positive or negative depending on the dissimilarity of properties adjacent to or farther from the slip zone, the direction of propagation and the sense of slip.
[40] As an example, consider material 1 (or plus) to be slightly more compliant and to have a slightly lower shear wave velocity than material 2 (or minus). In particular, c s1 = 0.90c s2 and c d1 /c s1 = c d2 /c s2 = ffiffi ffi 3 p , corresponding to m 1 = 0.75m 2 , r 1 = 0.923r 2 and n 1 = n 2 = 0.25. For this choice of Poisson's ratio (here to be interpreted as the undrained value), the Rayleigh wave speed in each material is 0.92 times the respective shear wave velocity. The generalized Rayleigh wave velocity, at which m(c GR ) = 0, is c GR = 0.87c s2 for these choices. [41] Figure 12 also plots values for W/2m 2 in order to compare the magnitudes of the bimaterial and porous media effects on alteration of the normal stress. As discussed in Appendix B, the properties entering W pertain to the nearfault material. Consequently, we choose the shear moduli, m + and m À , 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 where w + = w À = w and m + = m À = m. Results are shown in Figure 12 for values of the ratio k + b + /k À b À equal to 1, 10, 5, 2 and 1, 0.5, 0.2, 0.1 and 0. Since m ? (c GR )/m = 0.204 exceeds c(c GR ) = 0.155, the sum of the latter two terms changes from positive to negative and causes the slight downturn in values of W(v)/2 as v approaches c GR . Values of k + b + /k À b > 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 + /k À b À ). 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 ? /m 2 as v approaches c GR suggests, however, that the effect of the elastic mismatch will dominate near this limit, at least if B is not too large (larger values of B increase W) and 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) applies with the sign of the vertical axis reversed and the values of k + b + /k À b À replaced by their reciprocals. Thus, in general, any of the terms due entering (44) may be positive or negative and the sign of the term due to near fault heterogeneity (first in (44)) may differ from that of terms due to elastic dissimilarity (sum of last two entering (44)). Therefore the sign and magnitude of the alteration of the effective normal stress depends not only on the elastic dissimilarity of the materials bounding the fault but also on their differences from the material very near the slip surface and the differences of the poroelastic properties of this Figure 11. Plot of c, equation (42), against v/c s2 up to v = c GR for m 1 = 0.75m 2 , r 1 = 0.923r 2 and n 1 = n 2 = 0.25 so that c s1 = 0.90c s2 , c d1 /c s1 = c d2 /c s2 = ffiffi ffi 3 p and c GR = 0.87c s2 . Figure 12. Plot of m*(v)/m 2 against v/c s2 for the same elastic mismatch as in Figure 11: m 1 = 0.75m 2 , r 1 = 0.923r 2 and n 1 = n 2 = 0.25 so that c s1 = 0.90c s2 , c d1 /c s1 = c d2 /c s2 = ffiffi ffi 3 p and c GR = 0.87c s2 . Remaining curves show W/2m 2 for different values of the ratio of the product of the near fault permeability and compressibility on the two sides of the fault, k + b + /k À b À . On both sides of the fault, the near fault shear modulus is taken to be the average of values farther from the fault, B = 0.6 and n = 0.25. material on the two sides of the zone of concentrated sliding.

Discussion
[43] We have calculated the pore pressure induced by slip propagation for a model that idealizes recent detailed studies of fault zone structure [Chester et al., 1993;Chester and Chester, 1998;Lockner et al., 2000;Wibberley and Shimamoto, 2003;Sulem et al., 2004;Noda and Shimamoto, 2005]. In particular, slip is modeled as occurring on a plane bounded by material with permeabilities and poroelastic properties that are different on each side of the slip plane and from the properties of the elastic material farther from the fault. The pore pressure change is the result of poroelastic deformation of the fault wall; changes due to inelastic porosity changes, dilation or compression, of the fault zone material would be in addition to these. Although the detailed calculations are carried out for the limiting case in which the compressive side of the slip zone is much more permeable than the extensile side, we have shown that they can be applied to the more elaborate model simply by modifying the effective value of the Skempton coefficient. In particular, by changing the sign of the effective Skempton's coefficient from positive to negative treats the case in which the extensile side of the fault is more permeable.
[44] Induced pore pressure by poroelastic compression discussed here is one of a number of mechanisms that have been suggested for dynamic weakening of slip resistance during earthquakes. These include thermal pressurization of pore fluid [Lachenbruch, 1980;Mase and Smith, 1987;Rudnicki, 2003a, 2003b;Garagash et al., 2005;Rice, 2006], flash heating of asperity contacts [Rice, 1999;Tullis and Goldsby, 2003;Rice, 2006] and others [Sibson, 1975;Spray, 1993Spray, , 1995Goldsby and Tullis, 2002;Chambon et al., 2002]. For the most part, these require rapid slip to generate heat sufficiently rapidly and relatively large slip to generate sufficiently high temperature (although Segall and Rice [2006] have shown that shear heating can be significant toward the end of the nucleation period, before slip velocities become seismic). Although the mechanism discussed here increases in magnitude with increasing velocity of propagation, it is also operative at small slip and at low velocities. Consequently, it may be a factor in allowing sufficient slip to occur long enough for other mechanisms to come into play or for preventing incipient slip from progressing, depending on the sign of the induced pore pressure change.
[45] The model provides a more general framework for considering the effects of material heterogeneities perpendicular to the fault on alterations of the effective normal stress. Previous studies [Weertman, 1980;Andrews and Ben-Zion, 1997;Harris and Day, 1997;Cochard and Rice, 2000;Ben-Zion, 2001;Xia et al., 2005] have focused on the alteration of normal stress induced by inhomogeneous slip at the interface between elastic solids with different properties. The calculations here show that alterations of comparable magnitude in the effective normal stress can result from pore pressure changes induced by heterogeneous properties. More specifically, pore pressure changes result from differences in the permeability and poroelastic properties of the material on the two sides of the slip zone and from differences in the properties of this near fault material from that farther from the fault. Although parameter values are uncertain, a comparison of the magnitude of the two effects suggests that the effect due to nonuniform slip between different elastic solids may dominate as the rupture velocity approaches the generalized Rayleigh speed for the bimaterial, at least if the effective Skempton's coefficient is not too large, and the ratio of permeabilities is neither very large nor very small.
[46] Even when the effect due to elastic dissimilarity of material away from the fault is dominant, pore pressure changes due to near fault heterogeneity may augment or counteract the bimaterial effect. The sign of both effects can be positive or negative depending on the heterogeneity and the direction of slip and propagation. For example, the normal compressive stress is reduced by nonuniform slip when the direction of propagation of the slip distribution is the same as the direction of the fault wall displacement in the slower material. This reduction might, however, be diminished by a increase in effective compressive stress, due to a decrease in pore pressure, if the extensile side of the fault is more permeable. Thus the net effect of slip on the effective normal stress will depend on the details of the properties, both elastic and poroelastic, of the material bounding the zone of concentrated slip.
[47] Here the slip zone is assumed to occur at the interface of the two materials. Fault zone studies [Chester et al., 1993;Chester and Chester, 1998;Lockner et al., 2000;Wibberley and Shimamoto, 2003;Sulem et al., 2004;Noda and Shimamoto, 2005] do show that the principal slip surface is very narrow, less than about 1 to 5 mm, and thus reasonably idealized as a planar discontinuity for some purposes. Such studies also show that the relatively impermeable fault core is a wider zone of 10 mm to hundreds of millimeters and bounded by a more permeable, damaged zone (grading to undamaged material farther from the fault). The model shown in the inset of Figure 1 is consistent with slip occurring at the boundary of these two zones. If only the effect of near fault heterogeneity is considered, and the slip zone is plausibly assumed to take a path of least resistance where the effective compressive normal stress is least, then the results here indicate that the slip zone will choose a path where the compressive side is more permeable. For the direction of slip (right lateral) and propagation (to the right) shown in Figure 1, this is consistent with the positive material being the less permeable ultracataclastic core and the negative material being the adjacent more permeable, damage layer. Of course, this simple picture could be complicated by a variety of other effects. Nevertheless, in a recent numerical study of the bimaterial effect, Brietzke and Ben-Zion [2006] found that when several slip surfaces were possible, the rupture tended to choose a material interface where the compressive normal stress was reduced. It is possible to speculate that when the pore pressure changes due to material heterogeneity are included, the rupture tends to choose the interface where the reduction of effective normal stress is greatest, though this is certainly an issue in need of further study.
[48] Another issue in need of further study is the effect of near fault damage induced by slip propagation. In contrast to the scenario of the previous paragraph, such damage may have an effect which, typically, acts oppositely to the reduction of compressive normal stress due to far field elastic dissimilarity, by inducing negative pore pressure changes (suctions) along the slip surface. If there is no or only very small far field material dissimilarity, then the suctions will dominate and cause an increase in the effective normal compression, and hence partial stabilization of the fault. This possibility arises because a variety of studies [Poliakov et al., 2002;Kame et al., 2003;Rice et al., 2005;Andrews, 2005;Ben-Zion and Shi, 2005] have shown that unless the direction of maximum principal stress is at an unusually shallow angle with the fault, say, less than $25 degrees, then stresses predicted near the rupture front are expected to cause Mohr-Coulomb failure and damage preferentially on the extensile side of the slipping plane. Such preference is supported by field evidence [Poliakov et al., 2002]. Ben-Zion and Shi [2005] have shown this same tendency for damage on the extensile side in simulations of rupture along an interface between elastically dissimilar materials. The spatial extent of the damaged zone is predicted to decrease with depth  but the preferred side for damage remains. Presumably, this damage would increase the permeability on the extensile side whereas, conversely, compression of the other side may reduce the permeability there. The results here indicate that this effect, unless compensated by decrease of total compressive stress due to far field material dissimilarity, would decrease the pore pressure and hence increase the effective compressive stress and the energy required to propagate the fault. Ben-Zion and Shi [2005] have shown that including the effects of damage induced by rupture can modify and partially stabilize some aspects of the bimaterial effect. Further study of the effect on damage on altering the near fault permeability structure is needed.

Conclusion
[49] Interaction of pore fluid with material heterogeneity near the slip zone that is representative of that observed in fault zones can affect rupture propagation. An increase of pore pressure that reduces the effective compressive stress and facilitates slip propagation occurs if the compressive side of the slip zone (modeled as a plane) is more permeable than the tensile side; conversely, a decrease of pore pressure that increases the effective compressive stress and inhibits slip propagation occurs if tensile side is more permeable. Although a more complex and realistic model will undoubtedly alter the details of the calculations here, the main conclusion that near fault heterogeneity affects rupture propagation is unlikely to change. Understanding of how the particular effect studied here, pore pressure changes due to heterogeneous poroelastic properties, interacts with a variety of other effects must await further modeling and observational studies.
When g(t) is substituted from (8), the integral in the term multiplying (s xy 0 À t r ) can be done using contour integration by means similar to those described by RSP. The resulting expression for M(z) is (16) in the text. The Im{M + (x)}, which is needed to obtain the pore pressure can be calculated numerically directly from (16) and is also given by where the change of variable t = ÀsR has been used in the integral.

A1. Stress Drop
[53] Substitution of the expression for g into (A6) gives a relation between the ratio of the driving stress (s xy 0 À t r ) to the cohesive zone stress drop (t p À t r ) and the scaled cohesive zone size R/L: The integral in the denominator can be evaluated by converting it to a contour integral around the branch cut ÀL x < 0 in the complex plane and the result is Using (A10) and the substitution p = Àt/R in the integral in the numerator of (A9) gives (22) of the text.

A2. Relative Slip Displacements
[54] The variation of the relative displacement can be written as Substituting for Ds xx (x, 0 + ) from (9a) and using the identity (with Poisson's ratio taken as the undrained value) give @d @x where F(v) is defined following (24). Integrating and noting that d = 0 at x = 0 give where [55] Evaluating d(x) from (A14) at x = ÀL gives the total, locked-in displacement that has accumulated at the trailing edge of the slip zone d T . This total displacement can, however, be obtained directly by the same arguments used by RSP and Poliakov et al. [2002]. They show that d T is proportional to A and is given by Using (A4) for A and substituting the expression for g(t) from (8) yield If the second integral is denoted I, then dI/dL is equal to (1/2 À e) times the integral in (A10). Using this result and (22) and combining the integrals make it possible to write A as where D(R/L; e) is given by (25). Substituting (A18) into (A16) gives (24) of the text. This expression for d T agrees with (A14) evaluated at x = ÀL.

Appendix B: Pore Pressure at the Fault Plane
[56] In the main text the fault has sometimes been described, for simplicity, as a completely impermeable plane of displacement discontinuity in a uniform material, with pore pressure on the compressed side entering the friction law. Although it is a reasonable approximation to assume uniform material properties at some distance away from the fault, detailed examinations of fault zones [Chester et al., 1993;Chester and Chester, 1998;Wibberley and Shimamoto, 2003] show that the fault walls are often bordered by materials that are different from each other and from the uniform material farther away. In this Appendix, we show the effect of these different near fault properties, including the actual finite, if small, permeabilities of the materials involved (leading to a continuous pore pressure variation across the fault plane), can be included by modifying a parameter of the analysis based on discontinuous pressure at an impermeable plane in a uniform material. In particular, we assume that these regions of different properties bordering the fault are thin enough that they can be considered to undergo a uniform x direction strain e xx + (t) on one side of the fault and an equal and opposite strain e xx À (t) = Àe xx + (t) on the other side (Figure 1, inset). Because those bordering zones are presumed to be thin (compared, say, to the along-strike length scales R and L), the strains will be essentially identical to those calculated along the fault walls for the uniform material model in the body of the paper. These fault-parallel strains are related to the slip rate _ d at a fixed position x and rupture velocity v by where the compressive side of the fault is assumed to be y > 0. (Because the solution is steady state 2e xx + = @d/@x = À(1/v)@d/@t and this expression is consistent with (A11).) The strains are calculated as in the main text, based on the solution for an impermeable fault plane in uniform material.
[57] The hydraulic diffusivity a hy of fault gouge at representative seismogenic depths is generally estimated as being in the range of 1 -10 mm 2 /s [Rice, 2006], and the duration t of slip at a point in large earthquakes is typically of the order of 1 s for every 1 m of slip [Heaton, 1990]. Thus the thickness of the region over which fluid diffusion smooths out the discontinuity of the impermeable fault model, of the order of a few times ffiffiffiffiffiffiffiffi a hy t p , will not generally be larger than a few tens of millimeters. Thus we are concerned here with the properties and poromechanical response of border regions of that order of thickness along the fault walls.
[58] Now consider a magnified view of such border regions along the fault plane, so that they appear as two half planes subjected to a uniform x direction strain e xx + (t) in y > 0 and an equal and opposite strain e xx À (t) = À e xx + (t) in y < 0. The two half-spaces are assumed to be deforming under plane strain conditions (e zz = 0) and are subjected to the same fault normal stress s yy , which remains constant during slip. The shear stress s xy does vary but that does not affect the pore pressure under these conditions, assuming that the border regions are isotropic, or are aleotropic with principal directions aligned with the x and y directions. What is the pore pressure p f (t) induced at the fault plane y = 0?
[59] To answer this question, first imagine that the plane y = 0 is completely impermeable. Then the pore pressure changes p 0 ± (t) from the ambient value are uniform in each half-space and given by where B 0 = BW/w. Note, however, from (B10) that W need not be positive, and thus B 0 may be of either sign, depending on material properties in the two border regions. With this interpretation of B as B 0 , the solution for a uniform material with an impermeable fault plane, given in the main text, applies as well for the more realistic model of this Appendix. The case B 0 > 0 corresponds to weakening the fault by induced pore pressure, and B 0 < 0 to strengthening by induced pore suction.
[62] As a simple case, assume that all the near fault properties, except for the permeabilities, are identical to those of the homogeneous material farther from the fault. Then B 0 reduces to Hence, when the permeability of the extensional side is much less than that of the compressional, k À ( k + , then B 0 = B. If k À /k + = 10 À2 , which might be representative of having ultracataclasite on the extensional side and a coarser gouge or densely cracked material from the damaged fault core on the compressional side, based on properties inferred from Lockner et al. [2000] and Wibberley and Shimamoto [2003], then B 0 ' 0.82B; if k À /k + = 10 À1 , B 0 ' 0.52B. So the effect of decreasing permeability of the part of the fault core on the extensional side is to reduce the effective value of Skempton's coefficient in the solution. If the extensional side is the more permeable, which is the case generally expected based on where Mohr-Coulomb plasticity and damage is expected to occur most extensively near the rupture front, then k À > k + , and then B 0 < 0 and suction rather than pressure is induced on the fault plane. [63] More generally, if all properties of the border regions differ from those farther away, so that (B13) does not apply, but if it is nevertheless the case that the extensional side is essentially impermeable compared to the compressive, k À /k + ! 0, then The same expression for B 0 results, except that it is preceded by a minus sign and all plus superscripts are changed to minus, in the case for which the compressive side is essentially impermeable compared to the extensional, k À /k + ! 1. The term in brackets involving Poisson's ratio varies by at most a factor of three. For n u + = 0.4 and n u = 0.2, this bracket is 1.55. Thus, in these cases with one side being essentially impermeable compared to the other, B 0 is roughly equal to the near fault Skempton's coefficient B + reduced by the ratio of the shear modulus near the fault to that farther away, or to ÀB À reduced by a similar factor. The effect is to reduce the magnitude of the effective value of B, although the reduction might be partly offset by an increase in the undrained Poisson's ratio in the near fault material.