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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. 0, 10.1029/2000JB000138, 2002 Published in Journal of Geophysical Research, vol. 107 (no. B2), cn: 2030, doi: 10.1029/2000JB000138, pp. ESE.2.1 - ESE.2.17, February 2002. Pore pressure and poroelasticity effects in Coulomb stress analysis of earthquake interactions Errata Corrige page attached; one item Massimo Cocco refers to an error in the Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy published version, not present here. James R. Rice Engineering Sciences and Geophysics, Harvard University, Cambridge, Massachusetts, USA Received 3 January 2001; revised 2 October 2001; accepted 7 October 2001; published XX Month 2002. [1] Pore pressure changes are rigorously included in Coulomb stress calculations for fault interaction studies. These are considered changes under undrained conditions for analyzing very short term postseismic response. The assumption that pore pressure is proportional to fault- normal stress leads to the widely used concept of an effective friction coefficient. We provide an exact expression for undrained fault zone pore pressure changes to evaluate the validity of that concept. A narrow fault zone is considered whose poroelastic parameters are different from those in the surrounding medium, which is assumed to be elastically isotropic. We use conditions for mechanical equilibrium of stress and geometric compatibility of strain to express the effective normal stress change within the fault as a weighted linear combination of mean stress and fault- normal stress changes in the surroundings. Pore pressure changes are determined by fault-normal stress changes when the shear modulus within the fault zone is significantly smaller than in the surroundings but by mean stress changes when the elastic mismatch is small. We also consider an anisotropic fault zone, introducing a Skempton tensor for pore pressure changes. If the anisotropy is extreme, such that fluid pressurization under constant stress would cause expansion only in the fault-normal direction, then the effective friction coefficient concept applies exactly. We finally consider moderately longer timescales than those for undrained response. A sufficiently permeable fault may come to local pressure equilibrium with its surroundings even while that surrounding region may still be undrained, leading to pore pressure change determined by mean stress changes in those surroundings. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics, 7260 Seismology: Theory and modeling, 7215 Seismology: Earthquake parameters; KEYWORDS: Fault interaction, fluid flow, poroelasticity, effective friction, crustal anisotropy 1. Introduction fault plane to compute Coulomb stress changes. In the framework of the Coulomb criterion, failure on a fault occurs when the applied [2] Earthquakes produce changes in the state of strain and stress increment, defined as stress in the volume surrounding the causative faults. Coseismic stress and strain changes caused by shear dislocations are CFF ¼t þ mðs þ pÞ; ð1Þ usually calculated using the numerical procedure proposed by Okada [1985, 1992]. This approach is based on the solution of the elastostatic equations in an elastic, isotropic homogeneous overcomes a stress threshold, where. t is the shear stress change half-space. The coseismic strain and stress fields can be (computed in the slip direction), s .is the fault-normal stress computed if the geometry and the slip distribution on the change (positive for extension), p .is the pore pressure change rupturing fault plane are known (see Okada [1992], Stein et within the fault, and m is the friction coefficient which ranges al. [1992], King et al. [1994], Stein [1999], and King and between 0.6 and 0.8 for most rocks [see Harris, 1998, and Cocco [2000], among several others). In the near field the references therein]. The quantities included in (1) should be induced coseismic stress consists both of a dynamic (transient) considered as functions of time. and a static (permanent) perturbation. The calculation of [4] Earthquakes perturb the state of stress of a crustal volume, dynamic stress changes requires the solution of the elastody- and they cause a variety of hydrologic phenomena [Scholz, 1990; namic equations. It implies that both shear and fault-normal Sibson, 1994; King and Muir-Wood, 1994; Roeloffs, 1996, 1998; stresses can vary as functions of times reaching the static Roeloffs and Quilty, 1997]. Some of these effects can be explained configuration after a few tens of seconds [Harris and Day, by the poroelastic response to the earthquake-induced strain field. 1993; Cotton and Coutant, 1997; Belardinelli et al., 1999, and Pore pressure changes modify the coseismic stress redistribution, references therein]. and for this reason they are included in the definition of the [3] Fault interaction is currently investigated by means of these Coulomb failure function (1). Because the coseismic stress changes analytical formulations and using the induced stress on a specified occur on a timescale that is too short to allow the loss or gain of pore fluid by diffusive transport (fluid flow), the pore pressure changes included in (1) on that timescale are associated with the Copyright 2002 by the American Geophysical Union. undrained response of the medium [Rice and Cleary, 1976]. Later, 0148-0227/02/2000JB000138$09.00 we discuss somewhat longer timescales for which the fault is no ESE X-1 ESE X-2 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS longer undrained. The undrained conditions are those for which pressure changes are time-dependent; therefore it is necessary to there is no fluid flow. From an analytical point of view the specify the timescale during which the poroelastic model is undrained response implies that the fluid mass content per unit applied. volume is constant (m = 0), but the pore pressure is altered. [8] This paper discusses the assumptions required to correctly Under these conditions the relationship between stress and strain include the pore pressure changes in Coulomb stress modeling for a fluid-infiltrated poroelastic material is equivalent to an and provides a more general expression for the effective normal ordinary elastic material with appropriate coefficients for the stress for different timescales. We start investigating the short- undrained conditions [Rice and Cleary, 1976; Roeloffs, 1996]. term postseismic period, in which both the fault zone and the [5] According to Rice and Cleary [1976] the pore pressure adjoining lithosphere respond under undrained conditions. In this change resulting from a change in stress under undrained con- case we first assume that the fault zone is a poroelastic isotropic ditions is given by medium, but we will also consider the effect of anisotropy within the fault zone. Then we study an intermediate timescale, skk which will exist for a sufficiently permeable fault, during which p ¼ B ; ð2Þ the fault core reaches a local pressure equilibrium with its 3 lithospheric surroundings, while the adjoining lithosphere is still responding as if it were undrained. We will not consider here where B is the Skempton coefficient [Skempton, 1954; Kuempel, longer timescales during which the transition from short-term 1991]. Rice and Cleary [1976], Roeloffs and Rudnicki [1985], and undrained response to long-term drained response takes place Roeloffs [1996] present a compilation of experimental determina- also in the surrounding lithosphere. tions of B indicating a range between 0.5 and 0.9. In Coulomb stress analysis [see Stein et al., 1992; Harris and Simpson, 1992; King et al., 1994; Harris, 1998, and references therein] it is assumed that for plausible fault zone rheologies the change in pore 2. Poroelastic Constitutive Relations pressure becomes proportional to the fault-normal stress: [9] The stress-strain relation for an ordinary isotropic linearly elastic solid can be expressed as ^ p ¼ Bs: ð3Þ l 2Geij ¼ sij skk dij ; ð5Þ This is certainly true if in the fault zone s11 = s22 = s33, so 3l þ 2G that skk/3 = s and (2) becomes (3) [see Simpson and Reasenberg, 1994; Harris, 1998]. By substituting (3) in (1), we where eij and sij are the strain and stress tensors, respectively; G obtain and l are the Lamé parameters (G is the rigidity) and dij is the Kronecker delta. Hooke’s law (5) can be rewritten using the CFF ¼tþm0 s; ð4Þ Poisson ratio v.. as v where m0 = m(1B̂) is the effective (or apparent) friction coefficient. 2Geij ¼ sij skk dij : ð6Þ Equation (4) is very common in the literature, and it has been 1þv widely used to calculate Coulomb stress changes [see Harris, 1998, and references therein]. A variety of values are used for these Because here we consider linear elasticity, these constitutive calculations: the friction coefficient m ranges between 0.6 and 0.8, relations must be applied to small stress-strain magnitudes. We while B ranges between 0.5 and 1 [Green and Wang, 1986; Hart, assume that they are valid in such isotropic form for coseismic 1994]. The resulting values for the effective friction coefficient stress-strain changes caused by shear dislocations, which are of range between 0.0 and 0.75 (0.4 has been used in many interest since we do not know the absolute value of the regional calculations by Stein et al. [1992] and King et al. [1994]). Several remote tectonic stress. studies have concluded that Coulomb stress modeling is only [10] Because compact rocks consisting of solid phase materials modestly dependent on the assumed value of the effective friction are not an appropriate model for the crust, we have to consider our coefficient [see King et al., 1994]. This result might depend on the medium as porous or cracked. The stress-strain relations for a choice of the poroelastic model (equation (2) or (3)) in Coulomb poroelastic medium are slightly different from (6) because they stress analyses. It is important to emphasize that the effective include the pore pressure term [Biot, 1941, 1956; Rice and Cleary, friction coefficient m’ is not a material property, but it depends on 1976]. According to Rice and Cleary [1976], these constitutive the ratios of stress changes in the medium [Byerlee, 1992; Hill et relations are al., 1993; Beeler et al., 2000]. [6] Several recent papers have focused attention on the corre- v 3ðvu vÞ lation between fault-normal stress changes and earthquake loca- 2Geij ¼ sij skk dij þ pdij ð7aÞ 1þv Bð1 þ vÞð1 þ vu Þ tions as well as seismicity rate changes [see Perfettini et al., 1999; Parsons et al., 1999; Cocco et al., 2000, and references therein]. However, it is still not well understood why these fault-normal 3r0 ðvu vÞ 3 m ¼ m0 þ skk þ p ; ð7bÞ stress changes should provide a better explanation of this correla- 2GBð1 þ vÞð1 þ vu Þ B tion than Coulomb stresses. Fluid flow (time-dependent) as well as the choice of the proper expression for p in Coulomb analyses where m0, r0 are the fluid mass content and the density measured might help to explain some aspects of this paradox. with respect to a reference state at which we take P = 0. Here v is [7] Beeler et al. [2000] pointed out that using the constant the Poisson ratio under drained conditions, whereas the term vu apparent friction model (equation (4)) in Coulomb analyses may represents the undrained Poisson ratio, which is a function of v, the provide a misleading view in estimating stress changes. They bulk modulus (K), and the Skempton coefficient (B) of the medium compare that model with an isotropic and homogeneous (same [see Rice and Cleary, 1976; Kuempel, 1991]. Equation (7b) shows properties within the fault as outside) poroelastic model, equations that for undrained conditions (m = 0), p = Bskk/3 yielding (1) and (2), and conclude that Coulomb failure stress shows (2) when pore pressure and mean stress changes are considered. considerable differences for different tectonic environments. It is Equation (7a) is equivalent to (6) for a poroelastic medium if v in important to emphasize that because of fluid flow the induced pore (6) is replaced by vu. In fact, using the relation (2) in (7a) to COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X-3 Surrounding Crust strain components are the same inside and outside the fault zone [Rice, 1992], namely, e110 = e11, e220 = e22, e120 = e12. Similarly, G, λu,Ku, νu conditions of mechanical equilibrium require that certain stress 3 1 components must be the same everywhere within the fault zone as in the nearby crust outside it, namely, s330 = s33, s310 = s31, s320 = s32. In other words, the equilibrium conditions on the problem h 2 restrict s33 .to continuity but leave the other normal components unrestricted. This means that the other stress components inside the fault zone may be different from those outside it. They are determined from mechanical constitutive relations. homogeneous [14] According to (5) we have the following relations: isotropic Fault Zone Parameters medium G', λ'u,K'u, ν'u 1 2lu 2ðe11 þ e22 Þ ¼ s11 þ s22 skk ð10aÞ G 3lu þ 2G Figure 1. Undrained poroelastic fault model. 1 2l0 2 e011 þ e022 ¼ 0 s011 þ s022 0 u 0 s0kk ; ð10bÞ describe undrained (e.g., coseismic) strain and stress changes, we G 3lu þ 2G get the following constitutive relation: where primes denote quantities inside the fault zone. Using the vu continuity conditions for the strain components appearing in (10a) 2Geij ¼ sij skk dij : ð8Þ 1 þ vu and (10b) and for the fault-normal stress, we can write Equation (8) is equivalent to (6) but now represents a poroelastic 1 lu þ 2G 1 l0u þ 2G 0 medium under undrained conditions. skk s33 ¼ 0 s s33 : ð11Þ G 3lu þ 2G G 3l0u þ 2G0 kk [11] The following relation [Rice and Cleary, 1976] relates the undrained Poisson ratio to the other poroelastic parameters: [15] For a slightly more concise notation, let M = l + 2G be the modulus for one-dimensional strain and recall that K = l + 2G/3. 3v þ Bð1 2vÞ 1 KKs Then (11) becomes vu ¼ ; ð9Þ 3 Bð1 2vÞ 1 KKs 1 Mu skk 1 Mu0 s0kk s33 ¼ 0 s 33 ; G Ku 3 G Ku0 3 where K is the bulk modulus of the saturated rock under drained conditions and Ks is a modulus which, for certain simple materials where the quantity Mu/Ku corresponds to (1 vu)/(1 + vu), which (uniform properties of solid phase in response to hydrostatic might be alternatively used in the following equations (with primes stressing, fully interconnected pore space), can be equated to the denoting the values within the fault zone). Solving the previous bulk modulus of the solid grains in the rock. We emphasize that if relation for skk0/3 within the fault zone, we get K = Ks, then vu = v. In general, the undrained bulk modulus is larger than the drained one, and Ks > Ku > K. According to (9) the s0kk Ku0 G0 Mu skk G G 0 undrained Poisson ratio is larger than the drained Poisson ratio (vu ¼ 0 þ s33 : ð12aÞ > v). The same is true for the Lamé parameter, lu > l. The rigidity, 3 Mu G Ku 3 G on the contrary, remains the same, Gu = G. These considerations suggest that the fault stiffness for undrained conditions is larger To emphasize that this applies for the stress changes caused by a than that for drained ones. nearby earthquake, we write 3. Pore Pressure Changes in an Undrained s0kk K 0 G0 Mu skk G G 0 ¼ u0 þ s33 ð12bÞ Poroelastic Fault Model 3 Mu G Ku 3 G [12] We investigate the stress conditions for a fault in a and understand the unprimed stress changes (sij) to be those poroelastic medium. The fault zone materials have different conventionally computed by elastic dislocation theory. For properties with respect to the surroundings. Let 1 and 2 represent Coulomb analysis we need to know the pore pressure changes the coordinate directions in the fault zone and 3 represent the induced in the fault zone, which we obtain by substituting (12b) coordinate direction perpendicular to the fault plane (Figure 1). in (2): The stress-strain relations for the medium are given by (8). We 0 interpret the elastic moduli here as moduli for undrained defor- 0 0 0 skk 0 0 Ku G Mu skk G G0 mation. We indicate with G0, lu0 , Ku0 , vu0 the Lamé and bulk p ¼ B ¼ B 0 þ s33 ; ð13Þ 3 Mu G Ku 3 G moduli and the Poisson ratio within the fault zone, while G, lu, Ku, vu denote the parameters in the surrounding crust. In the where B0 is the Skempton coefficient in that fault zone. Equation following, we use the stress-strain relation for a poroelastic (13) shows that induced pore pressure changes depend both on the medium in the form of (5), using the Lamé moduli. mean stress and the fault-normal stress changes. The relevant [13] Considering the conditions of mechanical equilibrium and effective normal stress change for Coulomb stress analysis is thus strain compatibility, there exist equality conditions for strain and stress components within the fault zone and outside it, in the cases 0 0 0 0 skk that we consider here, for which fault zone thickness is much less seff 33 ¼ s33 þ p ¼ s33 B than length scales over which stress and strain vary outside the 3 K 0 G0 Mu skk G G0 fault. We indicate with eij0 and sij0 .the strain and the stress tensors ¼ s33 B0 u0 þ s33 : ð14Þ within the fault zone. Kinematic compatibility implies that certain Mu G Ku 3 G ESE X-4 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS [16] Equations (13) and (14) give the relative weights of fault- [20] If the density contrast between the fault zone materials and normal and mean stress perturbations in determining the pore the surrounding crust is not very large (r r0), the relative pressure and Coulomb stress changes. In general, the pore pressure variation of G and M can be expressed as changes depend on both these quantities. In particular, the pore pressure changes are related to the fault-normal stress changes G G 0 rVS2 r0 VS0 2 VS 2 VS0 2 through the rigidity contrast between fault zone materials and the ¼ ; surrounding crust. G rVS 2 VS2 0 2 [17] We can recognize two limiting cases for (14). The first one G0 VS holds when G0 = G and the mean stress is the relevant quantity. ¼ ; ð19Þ G VS Thus (14) becomes 0 2 Mu0 VP ¼ : 0 Ku0 Mu skk Mu VP seff 33 ¼ s33 B ; ð15Þ Ku Mu0 3 [21] An opposite limiting case exists when the rigidity contrast the second limiting case is obtained when G G0, and (14) is negligible (G0 approximately equal to G), and therefore we have becomes 0 VS2 VS0 2 r0 r 0 Ku : seff 33 ¼ 1 B s33 : ð16Þ VS2 r0 Mu0 [18] Equations (15) and (16) express the effective normal stress [22] This latter case represents the limiting case (G0 approxi- changes for the case where pore pressure changes depend solely on mately equal to G) yielding (15), and mean stress perturbations are mean or fault-normal stress changes, respectively. Equation (16) is the only contribution to the effective normal stress changes. equivalent to the effective friction approach of (4) if the Skempton [23] In a first simplified model we assume that the fault zone is a parameter is given by B̂ = B0K0u/M0u. solid of the same lithology as the adjoining crust, densely fractured with an isotropic distribution of cracks saturated by fluids, while the crust is considered as a much less cracked but still saturated 4. Elastic Moduli in the Fault Zone Poissonian body (l = G). We consider that none of the crack walls can open or close toward one another when an isotropic stress is [19] Equation (13) relates pore pressure changes to fault- applied, so they produce no change in volume and hence no normal and mean stress changes through two factors, which alteration of K0u. We can thus observe that the bulk modulus K0u depend on the elastic parameters in the fault zone and in the is unaffected by the presence of saturated cracks, at least assuming surrounding crust. The variation of these elastic parameters is that their aspect ratio is much less than the ratio of liquid bulk reflected in the variation of P and S wave velocities. There- modulus to solid bulk modulus, as pointed out by O’Connell and fore information on shear wave velocity anomalies in the fault Budiansky [1974]. Assuming that the fault zone has the same zone might be used to constrain numerical values of the lithology as the surroundings, just much more cracked, implies that factors appearing in (13) and to discuss the two limiting K0u = Ku. This also means that r = r0, neglecting the crack space cases reported in (15) and (16). In particular, the following contributions to volume. In these conditions the crack walls can relations hold (assuming that measured P wave speeds corre- slide in shear, so that G is reduced (G0 < G), yielding (19). spond approximately to undrained response in the sense of [24] We use seismic evidence for P and S wave velocity poroelasticity): variations to infer possible values of the elastic moduli in the fault 2 2 zone. Studies of local crustal tomography provide evidence on the Mu VP Mu V 4 body wave velocity variations in fault zones. Although several ¼ ; Ku ¼ G 43 ¼ G P2 ; ð17Þ studies (mostly based on VP tomographic images) interpreted the G VS G VS 3 fault zone as a high-velocity body [Lees, 1990; Lees and Nich- olson, 1993; Zhao and Kanamori, 1993, 1995], many others have and the same relations with primes (K0u, M0u) indicate the moduli suggested the presence of fluids within the fault zone [Eberhart- inside the fault zone. This equation yields Phillips and Michael, 1993; Johnson and McEvilly, 1995; Thurber et al., 1997]. Zhao et al. [1996] and Zhao and Negishi [1998] Ku 4 VS 2 found evidence of low P and S wave velocities and high Poisson ¼1 ð18Þ Mu 3 VP ratio at the hypocenter of the 1995 Kobe earthquake. We remark that VS and the Poisson ratio. vu (or VP/VS) are much more for the quantity which appears in (14). The square of the S wave sensitive to fluids than VP [see also Eberhart-Phillips and Reyners, velocity anomaly of the fault zone with respect to the 1999]. surrounding crust is related to the density and rigidity of the [25] Studies on fault zone trapped waves yield more useful two media: constraints to the quantities defined in (17) and (19) because they have an optimal resolution of the inner structure of fault zones whose thickness can range between 20 and 400 m [Li et al., 1990, VS2 VS0 2 r0 G rG 0 1994]. Li and Leary [1990] have shown the fracture density and the ¼ ; VS2 r0 G S wave velocity model for the Oroville (California) fault zone as determined by body wave travel time modeling. They clearly show where r is the density. A similar relation holds for the that the fault zone corresponds to a reduction in shear wave modulus for one-dimensional strain M and the P wave velocity velocity (roughly 50%) and an increase of the crack density (up anomaly: to 0.75). Li et al. [1990, 1994] point out that the reduction in S wave velocity inside the fault zone ranges from 30 to 50%. According to Mooney and Ginzburg [1986] and Li and Leary VP2 VP0 2 r0 Mu rMu0 ¼ : [1990] the velocity structure of the Calaveras fault shows a P wave VP2 r0 Mu velocity reduction of nearly 30%. COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X-5 [26] Although our review of velocity models of fault zones is far estimated does not affect the relative importance of mean stress from complete, we use these representative values to estimate the and fault normal stress in (13) for p0; it only affects the factor K0u/ quantities defined above. According to the aforementioned studies Mu0 in front. and to relations (17), (18), and (19) if V S0 /VS ranges between 0.5 and 0.7, the rigidity ratio G’/G ranges between 0.25 and 0.5. This implies that the rigidity reduction (G G0)/G is between 0.75 and 5. Modeling Static Stress Changes 0.5. As expected, if the reduction in S wave velocity within the From Shear Dislocations fault zone is very large (50 – 70%) the ratio G0/G becomes much smaller than (G G0)/G. Moreover, if we assume that the reduction [32] In this section we aim to compare the shear, fault- in S wave velocity is larger than that in P wave, then the V P0 /V S0 normal, and mean static stress changes caused either by a ratio inside the fault zone is larger than the corresponding value in vertical strike-slip fault or a normal fault in an elastic homoge- the surrounding crust. We consider the simplified model described neous half-space. We use the three-dimensional (3-D) dislocation above, which yields K u0 = Ku (= 5G/3 in a Poissonian surrounding code developed by Nostro et al. [1997], which is based on crust), and then given G0/G, we can calculate M u0 : numerical representation provided by Okada [1985, 1992]. Again, our discussion here is directed to the short timescale for which the fault and its surroundings respond as if undrained. 4 4 G0 5 4 G0 Mu0 ¼ Ku0 þ G0 ¼ K u þ G¼ þ G: The elastic stress changes caused by shear dislocations illustrate 3 3 G 3 3 G the spatial variability and absolute values of the different terms in (1), (2), and (4) and allow a comparison between the [27] This results in Coulomb stress changes resulting from the application of (1) as opposed to (4). Figure 2a shows the shear, normal, and Ku0 Ku 5 G 5 Coulomb stress changes caused by a vertical strike-slip fault ¼ ¼ ¼ 0 : Mu0 Mu0 3 Mu0 5 þ 4 GG mapped both on a horizontal layer at 6 km depth as well as on a vertical cross section A-A’ perpendicular to the fault strike. Coulomb stress changes have been computed by means of (4) [28] According to this relation the ratio Ku0 /M u0 ranges between on secondary faults having the same orientation and mechanism 0.714 and 0.883 for G0/G between 0.5 and 0.25, while it is equal to of the causative fault and using a constant effective friction 0.556 for G0/G = 1. As an example, we provide a tentative estimate equal to 0.4 (corresponding to m = 0.75 and B = 0.47). King et of the two proportionality factors that appear in (13). We assume a al. [1994] have already discussed these stress patterns in detail; reduction in P and S wave velocities in the fault zone of 18% and here we only remark that positive stress changes can favor 50%, respectively. These assumptions yield failures on appropriately oriented planes. We also point out that the variability of the stress patterns on the horizontal maps, G G0 G0 Ku K0 where both s1 and s3 lie, is more evident for strike-slip faults ¼ 0:75; ¼ 0:25; ¼ 0:556; u0 ¼ 0:883: G G Mu Mu than on the vertical cross sections. Moreover, the amplitudes of fault-normal stress changes at depth are quite small. It is [29] Using these values, the constants which multiply the mean important to remark that the cross section is not taken in the and the fault-normal stress changes in (13) and (14) are 0.45 and middle of the rupturing fault because such a direction is nodal 0.75, respectively. The latter does not seem to be negligible at all, for fault-normal stress changes, as shown in the map view. and for this illustration, Figure 2b shows similar results for a normal fault dipping 70 to the east; we only show the E-W vertical cross section calculated in the middle of the fault. As expected for a normal fault, the skk p0 ¼ B0 0:375 þ 0:625s33 largest spatial variability of stress changes occurs on the vertical 3 plane used for the sections where both s1 and s3 lie. ¼ B0 ½0:125ðs11 þ s22 Þ þ 0:750s33 : [33] Figure 3 shows the mean stress changes caused by the vertical strike-slip fault as well as the stress changes along the fault [30] Thus the assumption that p0 is proportional only to skk/ plane directions (1 and 2 in Figure 1). The stress changes in the 3 does not seem to be strongly supported by observation of P and S direction perpendicular to the fault plane (fault-normal stress) are wave velocities, but at the same time, neglecting this term may be shown in Figure 2a (middle). It emerges from these calculations justifiable only if the reduction in S wave velocity is much larger that the three diagonal terms of coseismic stress changes (sii) are than 50%, at least under the conditions assumed in the simplified substantially different in amplitude. In particular, the largest model considered here. This model can be reasonable for faults that amplitudes are found for the component oriented along the slip have experienced little slip, but it might be not reliable for a direction and that perpendicular to the fault plane, respectively. relatively mature fault zone. In this latter case, the presence of a Similar results have been obtained for a normal fault. The stress fault gouge with a different porosity, and possibly fluid-altered change is maximum in the direction of slip; this means s11 for a composition, with respect to the host lithology might be more strike slip fault. This result is also evident at depth, as shown in the properly represented by a density contrast [Mooney and Ginzburg, vertical cross sections in Figure 3. This implies that the condition 1986], so that r0 < r. Even in this case, we can show that the change s11 = s22 = s33, which leads to (4), is not satisfied in the in the one-dimensional strain modulus M is larger than the change volume surrounding the fault; moreover, the spatial variations of in density. According to Mooney and Ginzburg [1986] we can these stress components are quite different. This observation assume that VP/VP = Fr/r, with F 1, and write further supports the conclusion that (3) must be justified by different considerations, rather than assuming s11 = s22 = s33. VP 1 Mu0 r Mu0 r ¼ ¼ ð1 þ 2F Þ : [34] In Figure 4 we show the Coulomb stress changes calculated VP 2 Mu0 r Mu0 r both by (4) (the constant effective friction model) and by (1) and (2) (the isotropic model, see Beeler et al. [2000]). The two maps on [31] This gives a greater change in modulus M0u rather than in r the top of Figure 4 represent the Coulomb stress changes computed so that, approximately, neglect of r .changes in our estimate of using the same values of friction and Skempton parameters (0.75 modulus changes, as in (19), is still valid. Note that for a given G0/ and 0.47, respectively). In Figure 5 we show a similar comparison G, a different reduction from Mu to M0u than what we have for a normal fault in vertical cross section. It emerges from these ESE X-6 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS Figure 2. Shear, normal, and Coulomb stress changes caused by (a) a vertical strike-slip fault and (b) a 70 dipping normal fault calculated using the constant effective friction model of equation (4). The amount of slip on the rupturing fault is 50 cm. Coulomb stress changes have been computed on secondary fault planes having the same geometry and mechanisms as the causative faults. For all these calculations, m0 = 0.4, which corresponds to m = 0.75 and B = 0.47. The vertical cross section for the normal fault case shown in Figure 2b is taken in the middle of the causative fault. COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X-7 Figure 3. Map view and vertical cross section of mean stress changes (skk/3) and induced stress perturbations for the two isotropic components oriented along the fault directions (1 and 2 of Figure 1) caused by a vertical strike-slip fault as shown in Figure 2a. ESE X-8 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS Figure 4. Coulomb stress changes at 6 km depth caused by a vertical strike-slip fault computed with the constant apparent friction model (equation (4)) and the isotropic friction model (equations (1) and (2)). For this latter model we show the calculations using different values of the friction and Skempton coefficients. COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X-9 Figure 5. Same calculations as Figure 4 shown in a vertical cross section (as in Figure 2b) for a 70 dipping normal fault. ESE X - 10 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS calculations that the equation adopted for computing Coulomb stress affects the resulting spatial patterns. Such a result has been already discussed by Beeler et al. [2000], who also quantified the amount of such variations. They concluded that these two pore pressure models yield considerable differences in the calculated Coulomb stress changes for reverse, normal, and strike-slip faults. They also suggest that the use of the constant effective friction model (equation (4)) could lead to errors in estimating coseismic stress changes. The calculated Coulomb stress changes also depend on the assumed value of the Skempton parameter B [see also Beeler et al., 2000]. In Figures 4 and 5 we show the results of calculations using three different values of B between 0.2 and 1. As expected, increasing B increases the Coulomb stress changes in the off-fault lobes. This emphasizes the role of pore fluid pressure in earthquake failure [see, e.g., Segall and Rice, 1995] but also brings up the question of which is the most appropriate way to represent coseismic pore pressure changes. The differences among the Coulomb stress changes shown in Figures 4 and 5 can be as large as several bars. [35] To summarize, we have calculated the ratio between the mean stress (changed in sign: skk/3) and the fault-normal stress changes s(= s33)(see Figure 6a). Figure 6a shows that there is a region along the fault strike direction where this ratio is highly variable. Outside this region, on the two opposite sides of the fault, this ratio is negative and smaller than unity: This means that the mean stress and the fault-normal stress changes have the same sign but the latter is larger than the former. On the contrary, in the area where the ratio is positive they have opposite sign. A unitary value for this ratio would imply that p0 = B0s. The strike direction and that perpendicular to it are nodal for both these stress changes; thus, in these zones the amplitudes of both mean stress and fault- normal stress changes are very small (see Figures 2a and 3). In the off-fault lobes, where both mean and fault-normal stress change amplitudes are relevant, this ratio is negative and smaller than unity. This is quite evident in Figure 6b, where we plot the difference between normal and mean stress changes. These two terms differ mostly at the ends of the slipped zone. The implication of these calculations is that the effective friction coefficient is not constant in the volume surrounding the causative fault. This is expected, since from its definition it is a function of ratios of all the fault-normal stress changes to one another in the medium. [36] The considerations discussed above point out that in a poroelastic isotropic medium the relation used to compute pore pressure changes affects the calculation of Coulomb stress changes. Effective normal stress depends on both mean stress and fault- normal stress changes, and the proportionality factors do not vanish, except in very special situations that might not be realistic for actual fault zones. Moreover, the amplitudes of these stress changes show different spatial patterns, depending on the faulting mechanism [see also Beeler et al., 2000]. Therefore, in order to choose the appropriate equation to compute Coulomb stress it is necessary to consider more complex situations and different properties for the fault zone materials. 6. Effects of Anisotropy Within the Fault Zone [37] The results discussed above have been obtained under the assumption that the fault zone materials are isotropic, i.e., that they are permeated by an isotropic distribution of cracks that are saturated by fluids. However, anisotropy within the fault zone caused by aligned fractures may lead to different conclusions concerning the proportionality between pore pressure and fault- Figure 6. (a) Spatial pattern of the ratio between mean stress normal stress changes. Here we examine the effect of an aniso- perturbations (changed in sign: skk/3) and fault-normal stress tropic distribution of cracks within the fault zone, and we derive an changes (s = s33) for a vertical strike slip fault. (b) Spatial alternative formulation for the pore pressure changes for undrained pattern of the difference between normal and mean stress changes deformation. Because we are interested here in the undrained (skk/3). Worthy of note is the amplitude of such a difference. response of the medium, we do not discuss the anisotropy of permeability of the fault zone materials. COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X - 11 [38] In the general case, possibly anisotropic, equation (2), Hence, for that type of anisotropy of pore space, which may be which introduces the Skempton coefficient, must be generalized appropriate for a fault zone, it is correct to use the simplified to the statement that a pore pressure change concept that induced pore pressure under undrained conditions is determined solely by the change in fault-normal stress. In other sij words, if the anisotropy of the fault zone material is so extreme that p ¼ Bij ð20Þ when extracted from the fault and held at constant stress while fluid 3 is pumped into it, it expands only in the 3 direction, then p0 would is induced by application of stress changes sij under undrained be determined solely by s033 = s33, and then the m0 concept would conditions. The set of coefficients Bij constitute what we propose to apply exactly with B̂; = B033/3 in equation (3). call a Skempton tensor. They reduce, of course, to Bij = Bdij in the [43] It might be interesting to know how much anisotropy is isotropic case. If we regard P as a function of the set of stresses [s] needed to justify the effective friction concept in the way just and of the fluid mass m, per unit volume of reference state, discussed. Recent papers on fault zone trapped waves [Leary et al., contained in the porous material, that is, p = p([s], m), then Bij 1987; Zhao and Mizuno, 1999] have shown very good quality data, satisfy but unfortunately, there is still no answer to this question. Zhao and Mizuno [1999] found a crack density distribution for the 1995 Kobe (Japan) earthquake that is smaller (0.2) than that expected for Bij ¼ 3 @pð½s; m Þ=@sij : ð21Þ a fracture zone (0.6 – 0.75). It is important to point out that the presence of anisotropy within the fault zone may justify variations [39] We show in Appendix A that an alternative, and instructive, of shear wave velocity of 50% and larger. Future observations are interpretation can be obtained for that partial derivative once we needed to shed light on this problem; they will be helpful to recognize [Rice and Cleary, 1976] that sijdeij + pd(m/r) must be a reconstruct the inner structure and mechanical properties of fault perfect differential, where r is the density of the pore fluid zone materials. (conceptually in a reservoir of pure fluid at local equilibrium with the porous medium). Here it may be noted that m/r is the fluid volume fraction (fluid volume per unit of reference state volume of 7. Short Time Pore Pressure Equilibrium Between the porous material) in the case considered, when all pore space is Fault Zone and Adjoining Rock Mass connected and fluid-infiltrated. The mass of fluid per unit reference [44] We have focused thus far on the postseismic period, in state volume is m and r .is the density of pore fluid at pressure p, which the fault zone behaves as if it is undrained. However, if the and we assume r = r( p). The differential form sums the work of fault zone is moderately thin and has some permeability, then it is stresses in moving the boundaries of an element of the porous reasonable to expect that on what is also a relatively short time- material and the work of pore pressure in enlarging the boundaries scale, the fault zone will act as if it were locally drained and reach of the pore space. Together the terms constitute the change dU in pressure equilibrium with its surroundings, so that p0 evolves the strain energy U of the solid phase, which must be a function of toward p. p itself will be time-dependent because of the pore state. As developed in Appendix A, this is equivalent to the fluid fluxes set up by the gradients in the coseismically induced familiar notion from the thermodynamics of mixtures that sijdeij pore pressure field, but unless the fault zone is very thick and/or is + m̂dm sijdeij + m̂dm is a perfect differential, where m̂ is the very impermeable compared to its surroundings, that variation of P chemical potential of the pore fluid. in the adjoining rock will have a much longer timescale than for [40] By either route, the existence of the perfect differentials local drained response of the fault. In such cases it is reasonable to implies a Maxwell reciprocal relationship, which is shown in expect that p0 will have relaxed to p well before p itself has Appendix A to give the alternative interpretation of Bij as relaxed much from its undrained value just after the earthquake stress change. So, on such short but not extremely short timescale Bij ¼ 3r @eij ð½s; m Þ=@m : ð22Þ in which the fault acts as drained, but its surroundings remain undrained, we get p0 p and thus p0 is proportional to the The derivative corresponds to the change as fluid mass is pumped mean stress change outside the fault zone, p0 skk/3. into the porous material under conditions for which all of the [45] In order to model this behavior in a simple way, we stresses skl are held fixed. consider a fault zone of thickness h (see Figure 1) having a [41] Thus, for example, in the isotropic case B/(3r) is the uniform permeability k0, fluid viscosity h0, and storage modulus increase of each extensional strain per unit of increase dm of fluid N0. The surrounding crust is modeled as a pair of semi-infinite mass pumped into the material, under conditions for which the total domains with corresponding parameters N, k, h. The storage stresses are held constant. Equivalently, B/3 is the increase of each modulus is just the inverse of the storage coefficient (called S by extensional strain per unit increase dm/r of fluid volume pumped in Wang [2000]), in response to pore pressure changes, for one- under constant stresses. dimensional straining under constant stress in the straining [42] Let suppose that within the fault zone there is a highly direction. See Appendix B for its precise definition and expres- anisotropic distribution of cracks or flattened pores, lying so that sion in terms of moduli already introduced. As shown in Figure their long directions are approximately parallel to the fault plane. 1, the axis 3 is perpendicular to the fault. The diffusivity inside We isolate a sample of material of the fault and subject it to its in the fault zone (c0) and in the surrounding medium (c) can be situ stress state s0kl. Then, holding those stresses constant, we respectively defined as pump an increment dm0 of fluid mass into the porous material. In that case, because of the assumed orientations of the flattened c0 ¼ k 0 N 0 =h c ¼ kN =h; pores we would expect the fault-parallel strain increments de011 and de022 to be much smaller than the fault-normal component which depend on the viscosity, the permeability, and the storage de033 because it is the latter which would be primarily influenced modulus (the increase of fluid mass content when the pressure by fluid injection into the fault parallel-crack and pore space. varies under one-dimensional strain conditions) of the fault and of That means B033 is much larger than the other components of B0ij, the surrounding crust. and therefore that [46] By solving the one-dimensional consolidation problem for a layer of one porous medium within an effectively infinite outer ds0ij ds0 ds33 one, we model the evolution of the pore pressure within the fault dp0 ¼ B0ij B033 33 ¼ B033 : ð23Þ 3 3 3 zone toward its longer time limit p. The pore pressure inside the ESE X - 12 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS fault zone (p0) is a function of time and position in the fault zone, A) pore pressure evolution but for simplicity, we solve the problem at the center of the fault short timescale 2 2 zone (Appendix B). However, in the following, we will derive an ∆ p ′(t) − ∆p td = h + h  N 2 analytical expression for the pore pressure at any point inside the + ∆ p ′(0 ) − ∆p 8 c ′ 8c  N ′  fault zone. Our analysis of short-time undrained response shows that at time t = 0+ the fault zone has the pore pressure change p0 1 (given by equation (13)), whereas the surrounding crust has the c′  N  2 r= change p = Bskk/3. We determine the subsequent temporal c  N′  evolution by finding the difference between the pore pressure 0.8 inside and outside the fault zone, normalized by its value just after the time of the main shock (t = 0+): 0.6 r =1 r =∞ 0 4 P~ ¼ p ðt Þ p : ð24Þ 0.4 ∞ p0 ð0þ Þ p r =0 0 0 0.2 0.01 [47] Here the notation p (t) denotes p (z, t)jz=0,i.e.,the pore 0.11 pressure change at the center of the fault zone. In AppendixB 1 we solve simple diffusion equations using the Laplace transform 0 and its Bromwich inversion to find an expression for the 0 0.5 1 t / td 1.5 2 variable defined in (24)that depends on the parameter r, which is the ratio of diffusivities and storage moduli inside and outside B) pore pressure evolution , the fault zone: long timescale 2 2 0. 1 ∆p ′(t) − ∆ p h + h  N 2 k0 0 td = hAN c0 N 2 ∆ p ′(0 + ) − ∆ p 8c ′ 8c  N ′  @ r¼ . ¼ : k N0 c N0 r =0 h 1 0.01 0.11 [48] Once the solution is found (equation(B2) in Appendix B), 0.8 1 we estimate what parameters control the timescale over which the c′  N 2 4 fault zone reaches pressure equilibrium with its surroundings. In r= c  N ′ ∞ Appendix B we show that the time at which the pore pressure at the 0.6 center of the fault has evolved approximately half way toward its longer time limit of p is given by 0.4 h2 h2 N 2 h2 td ¼ 0 þ ¼ 0 ð1 þ rÞ: ð25Þ 0.2 8c 8c N 0 8c [49] This time constant is the sum of a characteristic drainage 0 time for the fault core plus a characteristic time for the crust. Two 0 5 10 t / td 15 20 limit cases exist: whenr=0, the crust is much more permeable than the fault zone, and more interesting ly, when r!1, the fault Figure 7. Temporal evolution of normalized pore pressure zone is much more permeable than the surrounding crust. In the alteration at a (a) short and (b) long timescales for different former case the time constant is simply td=h2/8c0, while the latter values of the parameter r(see text for its definition). The time condition yields variable is normalized using the characteristic time for pore pressure equilibrium, while the pore pressure alteration is h2 N 2 normalized at its initial value at the instant of application of td ¼ : the induced stress. 8c N 0 [50] In Figure 7 we show the temporal evolution of the normalized pore pressure alteration (defined in (24)) for short representing a slow temporal decay as t1/2. This slow asymptotic and long timescales and for different values of the parameter r. decay rate is independent of the permeability within the fault Figure 7 shows that the solution for r4 is almost identical to zone, although the time at which that asymptotic expression that obtained for r = 1. The pore pressure alteration at the fault becomes accurate does depend on fault permeability, since it center is halfway toward its longer time limit after a time which enters in the ratio r. Infact, that asymptotic form, which is varies from 0.75 td when r = 0, to 1.20 td when r = 1. This instructively written as property is what motivated our definition of td. We may also note that for all cases except r = 0, there is a slow evolution at long N p0 ð0þ Þ p p0 ðt Þ p ! pffiffiffiffiffiffiffi h; times, and thus the process is not readily characterized solely by 2 pct N0 td. Indeed, the asymptotic evaluation of the solution (B2) for large t shows that is valid in the generalized form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p0 ðt Þ p 2rtd h N N þh=2 p0 ðz; 0þ Þ p ! ¼ pffiffiffiffiffiffiffi ; p0 ðt Þ p ! pffiffiffiffiffiffiffi dz ð26Þ p0 ð0þ Þ p pð1 þ rÞt 2 pct N 0 2 pct h=2 N 0 ð zÞ COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X - 13 when we do not assume that the poroelastic material properties are Table 1. Characteristic Times for Local Pressure Equilibrium uniform within the fault zone. Rather, in (26) we assume that all Fault Thickness h, m Permeability material properties vary with position z within the fault zone, such asN 0 = N 0(z)but are uniform outside it (atjzj>h/2). Here k = 1018 m2 k = 1021 m2 4 p = Bskk/3 as before in the adjoining crust, and (26)applies 0.001 2.5 10 s 0.25 s on a sufficiently long timescale that local pore pressure equilibrium 0.01 2.5 102 s 25.0 s is achieved within the fault zone, with p0(t) denoting that value. 0.1 2.5 s 42 min Also, p0(z, 0+) is the pore pressure change that would have been 1. 4.2 min 2.9 days induced at position z (which coincides with the axis 3 in Figure 1) 10 6.9 hours 9.7 months 100 29 days 80 years within the fault zone in undrained response to the stress change. It is 1000 8.0 years 80 centuries given by our earlier expression for p0 as a linear combination of skk/3 and s33 but now written as Ku0 ð zÞ G0 ðzÞ Mu skk G G0 ð zÞ p0 ðz; 0þ Þ ¼ B0 ð zÞ þ s33 and those of its surroundings. We have shown that if the rigidity Mu0 ð zÞ G Ku 3 G inside the fault zone is much smaller than that in the surrounding crust, which should correspond to a large reduction of S wave This allows, for example, for a fault core and bordering zone with velocity within the fault zone, the pore pressure is primarily damage degrading gradually toward that appropriate for the controlled by the fault-normal stress changes and the definition surrounding crust. of the effective friction coefficient given in literature is tenable. [51] It is interesting to provide a tentative evaluation of the However, we emphasize that if the fault zone is an isotropic characteristic time values for local pressure equilibrium. It poroelastic medium permeated by fluids, the fault-normal stress emerges from (25) that the characteristic time will be dominated changes are the most important factor controlling the pore pressure by the smallest value of diffusivity, c or c0, which effectively changes only when the reduction in S wave velocity is >50%. A means the smallest permeability, since it is reasonable to assume limiting case, for which only the mean stress (i.e., the first that the storage modulus and the fluid viscosity do not signifi- invariant) controls the pore pressure, is found for faults whose cantly change between the fault zone and the surrounding crust. rigidity is equal to that of the surrounding crust. However, such a In Table 1 we list the values of td = h2/8cmin, understanding for condition appears inconsistent with observations of fault zone cmin the smaller diffusivity value. We assume a fluid viscosity of structure resulting from seismic tomography and fault zone trap- 2 104 Pa s and a storage modulus of 100 GPa. We have ped-wave studies. These studies consistently show that body wave calculated the characteristic times for two values of permeability: velocities within the fault zone are different from those in the 1018 m2 and 1021 m2, respectively, and for fault thickness in surrounding crust. In particular, fault zone trapped-wave studies the interval 1 mm h 1 km. As expected, the resulting values indicate that shear wave velocities in fault zones are as much as of the characteristic time for local pressure equilibrium change 50% smaller than in their surroundings. If the fault zone materials from seconds to years. This suggests that it is really difficult to do not behave as in these extreme conditions, both the mean stress exclude a priori the contribution of time-dependent pore pressure and the fault-normal stress changes contribute together to the pore equilibrium in the analysis of stress redistribution. Moreover, a pressure changes for undrained deformation during such a short complex fault network will inevitably have segments at different timescale. stages of their relaxation from undrained conditions to local [54] Calculations of Coulomb stress changes caused by shear pressure equilibrium with the nearby materials. dislocations in an elastic isotropic half-space show that the choice of the pore pressure model influences the results significantly. In particular, we show that the use of a constant 8. Discussion and Concluding Remarks effective friction model (equations (3) and (4)) as opposed to an isotropic and homogeneous pore pressure model (equations [52] Postseismic stress redistribution is a time-dependent (1) and (2)) implies very different Coulomb stress changes [see process, and at short or intermediate timescales (from minutes also Beeler et al., 2000]. These stress changes also depends on to few years after a seismic event), fluid flow can be one of the the assumed values of friction and Skempton parameters. most important factors in contributing to this temporal variation [55] We also briefly investigated the effect of anisotropy in the of the stress perturbation [Nur and Booker, 1972; Hudnut et al., cracked region forming the fault core. In this case, the Skempton 1989; Noir et al., 1997]. In order to properly include a pore parameter becomes a tensor. However, it is plausible to assume that pressure model in Coulomb analysis through equation (1) it is the strain component normal to the fault is much larger than those necessary to choose the timescale during which the stress in the fault-parallel directions. In that case, the pore pressure changes are modeled as well as to make a few assumptions should be solely dependent on the fault-normal stress, and the on the material properties of the medium. In this study we have definition of the effective friction coefficient should be correct. investigated two different timescales. We have first focused our Thus we can conclude that at very short postseismic time periods in attention on the short-term postseismic period, in which both the which the fault zone obeys undrained conditions, the constant fault zone and the adjoining lithosphere respond under undrained effective friction model could be an acceptable approximation only conditions. Thus we neglect the alteration of pore pressure under quite extreme conditions, such as if the fault zone rigidity is caused by fluid flow. In this first configuration we discuss the <50% that of its surroundings or if the fault zone is strongly pore pressure changes both in an isotropic poroelastic medium anisotropic. This latter configuration would be approached if and in an anisotropic fault zone. porosity is dominated by an oriented distribution of cracks or [53] That first condition allows a comparison with most of the flattened pores aligned with their long directions subparallel to the Coulomb stress studies. We have derived an analytical expression fault plane. that relates pore pressure changes for undrained deformation within [56] We have also discussed an intermediate timescale, which the fault zone to mean and fault-normal stress changes for an will exist only for a sufficiently permeable fault that is locally isotropic poroelastic medium. Both these terms contribute to the drained and reaches pressure equilibrium with its surroundings. In variations of pore pressure caused by the stress redistribution this time period the adjoining lithosphere is still responding as if it process. Their relative weight in the derived equation depends on were effectively undrained. In this case, we assume that the fault the contrast between the elastic parameters inside the fault zone zone is moderately thin and has some permeability. During this ESE X - 14 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS relatively short timescale, p0 evolves toward p, the former where p0 is an unessential reference pressure, we may use the being time-dependent due to the pore fluid fluxes generated by property that d m̂ = dp/r to obtain gradients in the coseismically induced pore pressure field. This variation of P in the adjoining rock will be slower than the local m pm m drained response of the fault. Therefore, on such a short but not pd ¼d dp ðA3Þ r r r extremely short timescale, during which the fault acts as drained, but its surroundings do not, we get p0 p, and thus p0 is pm pm ¼d md ðm ^Þ ¼ d m ^m þ m ^dm: proportional to the mean stress changes. The characteristic time of r r this local pressure equilibrium depends on the permeability and the fault thickness. Thus (A1) can be transformed to another perfect differential [57] The transition from short-term undrained to drained which is well known in the thermodynamics of mixtures, response in the adjoining lithosphere occurs in general on a yet namely, longer timescale. This should always occur as time increases, except when the fault zone is hydrologically isolated from its surroundings. In these circumstances, the stress changes sij m sij deij þ m ^dm ¼ d U þ ^ mm p : ðA4Þ approach the values that would be calculated from elastic disloca- r tion theory using drained, rather than undrained, elastic moduli. The drained response is elastically less stiff than the undrained [61] While it is unessential for what follows, the expression response: K < Ku, v < vu, G = Gu. In general, we might expect that of (A2) for m̂ may be seen to be consistent with interpreting this transition from undrained to drained response modestly m̂dm as the total reversible work of extracting an element of reduces the stress changes. For a mode II shear crack, in a plane mass dm from a reservoir of fluid at a reference pressure and strain condition, the reduction scales as (1 vu)/(1 v). For fluid density ( p0, r0 = r(p0)), and inserting it (say, through a Westerly granite this value is close to 0.9; thus the stress reduction porous screen) into a porous medium at a place where the at longer time periods seems almost negligible. However, a pore pressure and fluid density are (p, r). We assume that complete recognition of this behavior during such longer timescale temperature is the same in the reservoir as in the place of is beyond the aims of the present study. insertion and calculate the work in three steps, as follows: (1) [58] The time dependence is included in the Coulomb failure work of withdrawal from reservoir, p0dm/r0 (note that dm/r0 function not only through the pore pressure changes p (equation is the volume withdrawn from the reservoir),R (2) work of (1)). In fact, poroelastic theory shows that there is a time changing density from r0 to r, which is dm r0rpd(1/r). (3) dependence of all the stress components sij (we have considered Work of inserting the fluid at the place where pressure is p, here only timescales for which those sij are effectively constant which is pdm/r (dm/r is the volume inserted). The sum is outside the fault zone). Nur and Booker [1972] pointed out that m̂dm, so that the time dependence of pore pressure changes can interact with seismicity explaining aftershocks, and Rice [1980] evaluated the Zr Zp resulting time-dependent postseismic shear stress history on a dm dm fault surface. Further observations are needed to image the inner mdm ¼ po ^ dm pd ð1=rÞþp ¼ dm ð1=rÞdp; ðA5Þ ro r structure of fault zones and therefore to verify the most appro- ro po priate pore pressure model. In absence of constraining evidence we cannot exclude any model for including pore pressure in which is consistent with the expression for m̂ in (A2). Coulomb failure. However, at least in a homogeneous and [62] By a final rearrangement, we obtain isotropic poroelastic medium, the influence on pore pressure of the mean stress is well established. eij dsij þ m ^dm ¼ dV ðA6Þ as a perfect differential, where V = U + m̂m p(m/r) sijeij. Appendix A. Induced Pore Pressure for Regarding V as a function of the set of stresses [s] and fluid mass Undrained Stressing of an Anisotropic m, V = V([s], m), it therefore follows that Medium and Interpretation of a Skempton Tensor @V ð½s ; mÞ @V ð½s ; mÞ ^¼ m ; eij ¼ : ðA7Þ [59] An increment of work (per unit volume of the reference @m @sij state) done on a poroelastic material, precisely on its solid phase, is given by sijdeij + pd(m/r)[e.g., Rice and Cleary, 1976], where m/r Recognizing that @ 2V([s], m)/@mdsij must be independent of the is the fluid volume fraction defined as in the text. We recall that m order of differentiation, the Maxwell reciprocal relation is the mass of fluid per unit reference state volume of porous material and r .is the density of pure fluid at pressure p, and we mð½s ; mÞ @^ @eij ð½s ; mÞ assume r = r(p). This increment of work must be a perfect ¼ ðA8Þ differential of a function of state (i.e., of the strain energy U of @sij @m the solid phase, or of its Helmholtz free energy, at the constant temperature conditions considered), and so must be valid. Its left side can be rewritten as sij deij þ pd ðm=rÞ ¼ dU : ðA1Þ mð½s ; mÞ @^ mð pÞ @pð½s ; mÞ 1 @pð½s ; mÞ @^ ¼ ¼ ; ðA9Þ @sij @p @sij r @sij [60] Introducing the ‘‘chemical potential’’ m̂ = m̂(p) by and so the Maxwell relation is equivalent to Zp 1 ^¼m m ^ð pÞ ¼ d^p; ðA2Þ @pð½s ; mÞ @eij ð½s ; mÞ rð ^pÞ ¼ rð pÞ : ðA10Þ p0 @sij @m COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS ESE X - 15 [63] We recognize that @p([s], m)/@sij as being a generalization since N = ch/k. Thus, from equation (17) of Rice and Cleary of the Skempton coefficient B, valid for the anisotropic case as [1976], but with their expressions in terms of Poisson ratios well, and make the definition rewritten in terms of moduli used earlier here, we have N = B2Ku2M/(Ku K)Mu. Bij @pð½s ; mÞ [66] Measuring P relative to its value at time t = 0 (just before ¼ ðA11Þ the earthquake), so that p = p0 in the fault zone and p outside in 3 @sij the crust at t = 0+, and letting to define a Skempton tensor, with property that p = Bsij/ Z1 3 under undrained conditions. (Of course, in the isotropic case, pðz; sÞ ¼ ^ pðz; sÞest dt B ij = Bdij .) The Maxwell relation then gives us an interpretation of, and alternative way of understanding, the 0 Skempton tensor as be the Laplace transform, the solution of the above equation set is @eij ð½s ; mÞ pffiffiffiffiffiffiffiffi Bij ¼ 3r : ðA12Þ dp0 ð0þ Þ ½p0 ð0þ Þ p cosh z s=c0 @m ^pðz; sÞ ¼ pffiffiffiffiffiffiffiffi pffiffi pffiffiffiffiffiffiffiffi zjzj < h=2 s s cosh ðh=2Þ s=c0 þ r sinh ðh=2Þ s=c0 Appendix B. Short-Term Pore Pressure pffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi Equilibrium Between Fault Zone ^pðz; sÞ ¼ p ½p0 ð0þ Þ p r sinh ðh=2Þ s=c0 exp ðjzj ðh=2ÞÞ s=c0 þ pffiffiffiffiffiffiffiffi pffiffi pffiffiffiffiffiffiffiffi jzj > h=2; and Surrounding Crust s 0 s cosh ðh=2Þ s=c þ r sinh ðh=2Þ s=c 0 [64] The fault is modeled as a zone of thickness h (see Figure 1), which has uniform permeability k0, fluid viscosity h0, and storage where now p0(0+) is the same as p0 of (13), and the parameter: modulus N0 (defined below). The surrounding crust is modeled as a 0 pair of semi-infinite domains with corresponding parameters k, h, k h0 N c0 N 2 N. Our analysis of short-time undrained response shows that at r¼ ¼ : time t = 0+ the fault has the pore pressure change p0 (which we k 0 c N0 N have calculated in (13) as B0 times a linear combination of s33/ h 3 and s33), whereas the surrounding crust has the change p = Bskk/3. [67] It is simplest to invert the transform solution for the pore [65] The poroelastic equations then allow us to model the pressure at the center of the fault zone, and we use the notation evolution of the pore pressure in the fault toward its longer time limit p = Bskk/3. Recognizing that only e33 and no other strain varies with time, and noting that s33 is uniform, the same p0 ðtÞ pð0; t Þ; inside and outside the fault, the problem is recognized as one of one-dimensional consolidation. The governing equations within so that the fault and the crustal domains, respectively, incorporating Darcy’s law and conservation of mass of the diffusing fluid, for t ^p0 ðsÞ ^ pð0; sÞ: > 0 take the forms of Then k 0 dpðz; tÞ 1 dpðz; tÞ ¼ 0 ; h=2 < z < h=2 ! dz h0 dz N dt p0 ðsÞ p=s 1 ^ 1 1 pffiffiffiffiffiffiffiffi pffiffi pffiffiffiffiffiffiffiffi : d dpðz; tÞ 1 dpðz; tÞ p0 ð0þ Þ p s cosh ðh=2Þ s=c0 þ r sinh ðh=2Þ s=c0 kh ¼ ; z > h=2; z < h=2; dz dz N dt The Bromwich inversion integral is then where z is the spatial coordinate in the 3 direction, perpendicular to the fault. These are homogeneous diffusion equations for p. The 0þ general consolidation equations instead involve a homogeneous Zþi1 p0 ðtÞ p 1 ^p0 ðsÞ p=s st diffusion equation for m [Rice and Cleary, 1976], not p, but reduce ¼ e ds: ðB1Þ p0 ð0þ Þ p 2pi p0 ð0þ Þ p to such an equation for P in the case of one-dimensional straining, 0þ i1 like here. The diffusivities c0 = k0N0/h0 and c = kN/h. Their solution will be even in z and must satisfy, for t > 0, Since the integrand vanishes rapidly enough as jsj ! 1 and has no poles (at least when r > 0) but has a branch cut along the negative þ real S axis, the inversion path can be distorted to run from 1 to 0 h h along the lower side of the cut and from 0 to 1 along the upper p ;t ¼ p ;t 2 2 side. We make the substitution s = 4c0x2/h2 in the inversion k 0 dpðz; t Þ k dpðz; t Þ integral, where x is real and nonnegative along the distorted ¼ h0 dz h z¼h dz z¼hþ inversion path, and note that 2 2 est ¼ exp 4c0 t=h2 ¼ exp ð1 þ rÞx2 t=2td ; The storage modulus N (inverse of the storage coefficient) is defined such that dm = r0dp/N is the increase in fluid mass content where when the pressure varies under conditions of one-dimensional strain, with s33 held constant. Expressions for it can be extracted h2 h2 N 2 from those for c in the sources mentioned on poroelasticity [Biot, td ¼ þ : 1941, 1956; Rice and Cleary, 1976; Kuempel, 1991; Wang, 2000], 8c0 8c N 0 ESE X - 16 COCCO AND RICE: POROELASTICITY EFFECTS IN COULOMB STRESS ANALYSIS We find from numerical evaluations that td gives the time at which Eberhart-Phillips, D., and A. J. Michael, Three-dimensional velocity struc- the pore pressure at the center of the fault has evolved ture, seismicity, and fault structure in the Parkfield region, central Cali- fornia, J. Geophys. Res., 98, 15,737 – 15,758, 1993. approximately halfway toward its longer time limit of p = Eberhart-Phillips, D., and M. Reyners, Plate interface properties in the Bskk/3. Hence northeast Hikurangi subduction zone, New Zealand, from converted seis- mic waves, Geophys. Res. Lett., 26, 2565 – 2568, 1999. pffiffi Z1 ! Green, D. H., and H. F. Wang, Fluid pressure response to undrained com- p0 ðt Þ p 2 r sinð xÞexp½ð1 þ rÞx2 t=2td pression in saturated sedimentary rock, Geophysics, 51, 948 – 956, 1986. ¼ dx: ðB2Þ Harris, R. A., Introduction to special section: Stress triggers, stress sha- p0 ð0þ Þ p p x cos2 ð xÞ þ r sin2 ð xÞ 0 dows, and implications for seismic hazard, J. Geophys. Res., 103, 24,347 – 24,358, 1998. Harris, R. A., and S. M. Day, Dynamics of fault interaction: parallel strike- The integral converges rapidly for t > 0 and can be evaluated by slip faults, J. Geophys. Res., 98, 4461 – 4472, 1993. standard numerical integration schemes, although the number of Harris, R. A., and R. W. Simpson, Changes in static stress on southern numerical subdivisions becomes very large when r is either very California faults after the 1992 Landers earthquake, Nature, 360, 251 – large or small compared to unity. Fortunately, the limit cases can be 254, 1992. treated separately. We find that when r = 0 (crust very much more Hart, D. J., Laboratory measurements of a complete set of poroelastic moduli for Berea sandstone and Indiana limestone, M. S. thesis, 44 permeable than the fault zone), the branch cut changes into a row pp., Univ. of Wis., Madison, 1994. of simple poles, giving Hill, D. P., et al., Seismicity remotely triggered by the magnitude 7.3 Landers, California, earthquake, Science, 260, 1617 – 1623, 1993. Hudnut, K. W., L. Seeber, and J. Pacheco, Cross-fault triggering in the p0 ðt Þ p 4X1 ð1Þn 2 2p t November 1987 Superstition Hills earthquake sequence, southern Cali- ¼ exp ð 2n þ 1Þ r ¼ 0; p0 ð0þ Þ p p n¼0 2n þ 1 8td fornia, Geophys. Res. Lett., 16, 199 – 202, 1989. Johnson, P. A., and T. V. McEvilly, Parkfield seismicity: Fluid driven?, J. Geophys. Res., 100, 12,937 – 12,950, 1995. in which case td = h2/8c0. That is just the classical Terzaghi one- King, G. C. P., and M. Cocco, Fault interaction by elastic stress changes: dimensional consolidation solution for a layer which is freely New clues from earthquake sequences, Adv. Geophys., 44, 1 – 38, 2000. drained at both sides. King, G. C. P., and R. Muir-Wood, The impact of earthquakes on fluids in the crust, Ann. Geofis., XXXVII(6), 1453 – 1460, 1994. p[68] Also, either direct treatment or use of the substitution y = King, G. C. P., R. S. Stein, and J. Lin, Static stress changes and the x r and taking the limit r ! 1 (fault zone very much more triggering of earthquakes, Bull. Seismol. Soc. Am., 84, 935 – 953, 1994. permeable than crust) gives Kuempel, H. J., Poroelasticity: Parameters reviewed, Geophys. J. Int., 105, 783 – 799, 1991. Z1 Leary, P. C., Y. G. Li, and K. Aki, Observation and modelling of fault-zone p0 ðt Þ p 2 expðy2 t=2td Þ fracture seismic anisotropy, I, P, SV and SH travel times, Geophys. J. ¼ dy Astron. Soc., 91, 461 – 484, 1987. p0 ð0þ Þ p p 1 þ y2 0 r ¼ 1; Lees, J. M., Tomographic P-wave velocity images of the Loma Prieta earthquake asperity, Geophys. Res. Lett., 17, 1433 – 1436, 1990. Lees, J. M., and C. E. Nicholson, Three-dimensional tomography of the 1992 in which case, southern California earthquake sequence, Geology, 21, 387 – 390, 1993. Li, Y. G., and P. C. Leary, Fault zone trapped seismic waves, Bull. Seismol. h2 N 2 Soc. Am., 80, 1245 – 1271, 1990. td ¼ : Li, Y. G., P. C. Leary, K. Aki, and P. Malin, Seismic trapped modes in the 8c N 0 Oroville and San Andreas fault zones, Science, 249, 763 – 766, 1990. Li, Y. G., J. Vidale, K. Aki, J. Marone, and W. K. Lee, Fine structure of the The above expressions have been plotted in Figure 7 for various Landers fault zone: Segmentation and the rupture process, Science, 265, values of r on short and long timescales. 367 – 380, 1994. Mooney, W. D., and A. Ginzburg, Seismic measurements of the internal properties of fault zones, Pure Appl. Geophys., 124, 141 – 157, 1986. Noir, J., E. Jacques, S. Békri, P. M. Adler, P. Tapponnier, and G. C. P. King, [69] Acknowledgments. J.R.R. wishes to acknowledge the support of Fluid flow triggered migration of events in the 1989 Dobi earthquake the NSF Geophysics Program and the USGS Earthquake Hazards Reduc- sequence of central Afar, Geophys. Res. Lett., 24, 2335 – 2338, 1997. tion Program and also of a Blaise Pascal International Research Chair from Nostro, C., M. Cocco, and M. E. Belardinelli, Static stress changes in the Foundation of École Normale Supérieure, Paris. M.C. wishes to extensional regimes: An application to southern Apennines (Italy), Bull. acknowledge the Institute de Physique du Globe de Paris for the hospitality Seismol. Soc. Am., 87, 234 – 248, 1997. during his stay in France. This study was partially supported by the Nur, A., and J. R. Booker, Aftershocks caused by pore fluid flow?, Science, European Commission contract ENV-4970528, Project Faust. We thank 175, 885 – 887, 1972. T. 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Res., 104, 20,183 – 20,202, 1999. dynamic stress during coseismic ruptures: Evidence for fault interaction Perfettini, H., R. Stein, R. Simpson, and M. Cocco, Stress transfer by the and earthquake triggering, J. Geophys. Res., 104, 14,925 – 14,945, 1999. M = 5.3, 5.4 Lake Elsnam earthquakes to the Loma Prieta fault: Biot, M. A., General theory of three-dimensional consolidation, Appl. Unclamping at the site of the peak 1989 slip, J. Geophys. Res., 104, Phys., 12, 155 – 164, 1941. 20,169 – 20,182, 1999. Biot, M. A., General solutions of the equations of elasticity and consolida- Rice, J. R., The mechanics of earthquake rupture, in Physics of the Earth tion for a porous material, J. Appl. Mech., 78, 91 – 98, 1956. Interior, edited by A. M. Dziewonski and E. Boschi, Proc. Int. Sch. Phys. Byerlee, J. D., The change in orientation of subsidiary shears near faults Enrico Fermi, 78, 555 – 649, 1980. containing pore fluid under high pressure, Tectonophysics, 211, 295 – Rice, J. 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Rice, Engineering Sciences and Geophysics, Harvard University, 29 southern San Andreas fault system caused by the 1992 magnitude = Oxford St, 224 Pierce Hall, Cambridge, MA 02138, USA. (rice@ 7.4 Landers earthquake, Science, 258, 1328 – 1332, 1992. esag.harvard.edu) ERRATA CORRIGE (this compilation: 4 October 2002) 1. James R. Rice's correct affiliation (first page, just below title) is: Department of Earth and Planetary Sciences and Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts. 2. There is a mistake in the equation included in Figures 4 and 5 for the isotropic poroelastic model. The correct equation is: ∆CFF = ∆τ + µ(∆σ n − Β∆σ kk / 3) . The figures were computed with the proper sign and are correct; only that equation is misprinted. 3. Paragraph [38], Equation (21) should be: Bij = 3∂p([σ ], m) / ∂σ ij (correcting the placement of the closing square bracket and eliminating the unnecessary curly brackets). 4. Paragraph [40], Equation (22) should be: Bij = 3ρ∂εij ([σ ], m) / ∂m (again, correcting the placement of the closing square bracket and eliminating the unnecessary curly brackets). 5. End of paragraph [44]: The correct relation is: ∆p′ = −Β∆σ kk / 3 (instead of ∆p′ ≈ −∆σ kk / 3). 6. Appendix A, Equation (A3): The expression following the last equal sign should be:  pm  d − µˆ m + µˆ dm  ρ  7. Appendix B, paragraph [65]: The upper case P which appears between the two sets of equations in that paragraph should instead be lower case p. 8. Appendix B, paragraph [65]: Last member of second set of equations has the lower case deltas within both parentheses. All four such deltas should be the partial derivative sign ( δ should be replaced by ∂ ). 9. Appendix B, paragraph [66]: The first term on the right of the second equation has a lower case delta in the numerator which should be an upper case delta ( δ should be replaced by ∆ ). JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2069, doi:10.1029/2002JB002319, 2003 Correction to ‘‘Pore pressure and poroelasticity effects in Coulomb stress analysis of earthquake interactions’’ by Massimo Cocco and James R. Rice Received 22 November 2002; published 4 February 2003. INDEX TERMS: 9900 Corrections; KEYWORDS: fault interaction, fluid flow, poroelasticity, effective friction, crustal anisotropy Citation: Cocco, M., and J. R. Rice, Correction to ‘‘Pore pressure and poroelasticity effects in Coulomb stress analysis of earthquake interactions’’ by Massimo Cocco and James R. Rice, J. Geophys. Res., 108(B2), 2069, doi:10.1029/2002JB002319, 2003. [1] In the paper ‘‘Pore pressure and poroelasticity effects 4. Paragraph [40], Equation (22) should be Bij = 3r@eij([s], in Coulomb stress analysis of earthquake interactions’’ by m)/@m (again, correcting the placement of the closing bracket Massimo Cocco and James R. Rice (Journal of Geophysical and eliminating the unnecessary curly brackets). Research, 107(B2), 2030, doi:10.1029/2000JB000138, 5. End of paragraph [44]: The correct relation is p0 = 2002), there are several corrections as follows: Bskk/3 (instead of p skk/3). 1. James R. Rice’s correct affiliation (first page, just 6. Appendix A, equation (A3): The expression following below title) should be Department of Earth and Planetary the last equals sign should be d pm r m ^ m þ m ^ dm. Sciences and Division of Engineering and Applied 7. Appendix B, paragraph [65]: The capital P that appears Sciences, Harvard University, Cambridge, Massachusetts. between the two sets of equations in that paragraph should 2. There is a mistake in the equation included in Figures instead be lowercase p. 4 and 5 for the isotropic poroelastic model. The correct 8. Appendix B, paragraph [65]: Last member of second set equation is CFF = t + m(sn Bskk/3). The figures of equations has the lower case deltas within both were computed with the proper sign and are correct, only parentheses. All four such deltas should be the partial the text is wrong. derivative sign (d should be replaced by @). 3. Paragraph [38], Equation (21) should be Bij = 3@p([s], 9. Appendix B, paragraph [66]: The first term on the right m)/@sij (correcting the placement of the closing bracket and of the second equation has a lowercase delta in the numerator eliminating the unnecessary curly brackets). which should be an capital delta (d should be replaced by ). Copyright 2003 by the American Geophysical Union. 0148-0227/03/2002JB002319$09.00 ESE 2-1