Figure 10.1

      As one of the best exposed tectonic-plate boundaries in the world, the San Andreas fault provides an excellent opportunity to study the forces causing interplate motion and the associated great earthquakes. Thus, there is considerable motivation, scientific, social, and economic, to understand the thermomechanics of the San Andreas fault system, which has been the subject of intensive studies for the past several decades.

      Although substantial progress has been made in unraveling the complex kinematics of the San Andreas fault system (Atwater, 1970; Minster and Jordan, 1984; Weldon and Humphries, 1986), efforts to determine the stresses that give rise to San Andreas fault slip, to date, have not led to anything resembling scientific consensus. The uncertainty results from widespread disagreement over the implications of different methods of assessing the stresses.
      The question of how much shear stress acts on the San Andreas fault to cause dextral slip began to acquire definition in 1968, when the first heat-flow data adjacent to the fault zone (fig. 10.1) were gathered and analyzed by Henyey (1968). Because these data did not reveal any anomalous heat flow near the major active faults of the San Andreas system, upper bounds of about 10 to 20 MPa on the average frictional stress resisting fault motion could be calculated (Brune and others, 1969; Henyey and Wasserburg, 1971). These upper bounds were taken as evidence confirming speculation on the low strength of the crust based on earthquake stress drops, almost invariably in the range 0.1-10 MPa (for example, Chinnery, 1964; Brune and Allen, 1967). At the same time, however, laboratory experiments indicated typical frictional strengths for precut rock samples of about 100 MPa under pressure and temperature conditions thought to obtain in the upper crust (Byerlee and Brace, 1968, 1969; Byerlee, 1970).
      Over the next several years, new heat-flow measurements supported the absence of any local heat-flow anomaly associated with the San Andreas fault (Lachenbruch and Sass, 1973) and thus augmented the position for low frictional fault strength. The recognition of a broad heat-flow high coincident with the Coast Ranges of California led Lachenbruch and Sass (1973) to suggest that partial decoupling at the base of the seismogenic part of the crust might account for both the weak fault (minimum in shear stress at the fault trace) and the broad thermal anomaly.
      Additional laboratory experiments on different rock types, and in conditions of higher temperature and confining pressure than had been obtained previously, continued to support high frictional strength in the top 15 to 20 km of the fault zone (Stesky and Brace, 1973). The experimental results are most simply characterized in terms of a coefficient of friction that varies little with rock type (Byerlee, 1978), slip rate, or slip history (Dieterich, 1979; Ruina, 1983). As emphasized by Brace and Kohlstedt (1980) and Kirby (1980), these results still indicate a high-strength upper crust.
      Beginning in the late 1970's, inplace stress measurements have provided another way to assess the stress acting on the San Andreas fault (Zoback and others, 1977), especially with the advent of stress measurements at depths approaching 1 km only a few kilometers distant from the fault (Zoback and others, 1980). If the observed depth gradient for the component of shear stress thought to act on the San Andreas fault could be extrapolated to the base of the seismogenic zone, as argued by McGarr and others (1982), then the corresponding frictional stress resisting fault motion is a factor of 3 greater than the upper bound from the heat-flow analyses, as presented most recently by Lachenbruch and Sass (1980).
      The most recent developments, if accepted at face value, could be construed as additional evidence favoring a low-strength San Andreas fault. Specifically, stress-direction indicators on either side of the fault have been interpreted to mean that there is almost no shear stress resolved on the fault plane, thus implying a very weak fault zone (Mount and Suppe, 1987; Zoback and others, 1987). If so, then the question regarding the strength of the fault would be answered, and the outstanding problem would be the equally vexing one of understanding the nature of a remarkably weak fault zone.
      This chapter is largely a review and commentary on the different approaches taken to estimate the tractions acting on the San Andreas fault. We restrict our attention to three main methods: (1) inferring stress from the fault's energy budget (thermal and kinetic), (2) inferring fault strength from laboratory measurements of the stresses needed to slide rocks past one another under pressure, and (3) inferring stress on the fault from observations of the crustal state of stress.

      In figure 10.2, an earthquake is viewed, according to Reid's (1910) rebound theory, as a strained patch of fault surface of area A that suddenly breaks, permitting points initially in contact to be displaced from one another by an average amount u. The breakage is like the sudden failure of an overloaded leaf spring. We are interested in the average shear stress acting parallel to the wall in the failed section of the fault surface. We denote its initial value by τ_l and its final value by τ₂. The inclined line in figure 10.3 represents the elastic unloading of the medium as the earthquake displacement increases to its final value u. The area under this line, which represents the total elastic energy released by the earthquake per unit area of faulted surface, can be written as

The energy E must supply the work E_a of generating seismic waves and the work E_r, converted to heat in overcoming frictional resisting forces. Thus,

where the question mark is a reminder (which we shall forget for the moment) that there may be other significant sinks of earthquake energy, such as the surface energy consumed in creating new fractures. We can now write

where τ_a, the "apparent stress" of seismology, is the portion of the earthquake stress allocated to the production of seismic waves, and τ_r is the average frictional resisting stress allocated to the production of heat. The individual areas represented by equations 3 and 4 are shown in figure 10.3 by contrasting patterns.

Figure 10.2 - Elastic-rebound theory (Reid, 1910), showing displacement near a strike-slip fault segment of area A before and after an earthquake with displacement u. Arrows along fault indicate direction of relative movement.

Figure 10.3 - Relation between resisting stress and displacement in the unloading elastic medium (inclined line) during an earthquake. As slip (u) increases, stress in the rock diminishes linearly from τ1 to τ2, with average value . Area under this line is total work expended per unit fault area; area (shaded) below curve of resisting stress (τr(u), with average value r) is energy dissipated per unit fault area (Er). Difference between total work expended and dissipated energy is work (done by apparent stress τa) that is available for seismic radiation (Ea, stippled area). Modified from a laboratory experiment on a large granite sample by Lockner and Okubo (1983).

      This interpretation of the areas in figure 10.3 is fairly general, as long as we define τ₁, τ₂, and τ_r, respectively, as the weighted averages of initial stress, final stress, and friction over the faulted surface, the weighting function being the local fault slip (see Savage and Wood, 1971; Lachenbruch and Sass, 1980). Combining equations 1 through 4 yields

which states that unless the question mark represents something important that we've neglected, the average earthquake stress is the sum of the apparent stress τ_a, to be estimated from seismic measurements of E_a (eq. 3), and the resisting stress τ_r, to be estimated from thermal measurements of frictional heat E_r (eq. 4).

      Seismologists (for example, Wyss and Brune, 1968; Savage and Wood, 1971; Wyss and Molnar, 1972) have defined apparent stress as

where η is the seismic efficiency, defined by

where equation 7b follows from 7a according to equations 2 through 5; that is, η is simply the fraction of the total energy release, or the fraction of the average earthquake stress, allocated to seismic radiation.
      To estimate τ_a, seismologists first determine the radiated energy E_a and the seismic moment M₀, defined as

where u is the average slip of an earthquake over a fault-surface area A, and G is the modulus of rigidity (Aki, 1966). Equations 1 and 6 through 8 then yield the simple relation

      A numerical estimate of τ_a can be obtained from equation 9 with the following commonly used formulas relating earthquake magnitude M to E_a or M₀ (Gutenberg and Richter, 1956; Hanks and Kanamori, 1979),

where M₀ and E_a are in ergs. Substitution of equations 10 in 9 yields

With G=3x10⁴ MPa, the value for τ_a is 2 MPa. Almost without exception, estimates of τ_a fall within the range 0-5 MPa, with no indication of any systematic dependence on either earthquake size or tectonic environment (Spottiswoode and McGarr, 1975; Fletcher and others, 1983; Boatwright and Choy, 1986). In short, 5 MPa appears to be a conservative upper bound to τ_a. Thus, the contribution of τ_a is small, and the average fault stress can be large only if the frictional resistance τ_r is large (eq. 5).
      If laboratory "earthquakes" are proper analogs of crustal earthquakes, which mayor may not be the case, then data for such events, including those illustrated in figure 10.3, indicate that τ_a is indeed small, only a tiny fraction of τ_r. By inducing unstable frictional failure (earthquakes) across a 2-m-long fault between slabs of granite 40 cm thick (Dieterich, 1981), Lockner and Okubo (1983) measured seismic efficiencies η for numerous events to conclude that η≈0.05. If this result were true also for natural earthquakes - a big "if" - then for a typical value τ_a of 2 MPa, the corresponding value of , from equation 6, would be 40 MPa, which, as will be seen, is nearly 3 times higher than the limit inferred from an analysis of heat-flow data (Lachenbruch and Sass, 1980).

      Unlike the energy of seismic waves, which permits an estimate of apparent stress τ_a for individual earthquakes from measurements at distant stations, the heat energy of individual earthquakes is not readily analyzable to estimate friction because it causes a measurable temperature rise only within a few meters of the earthquake source, a location inaccessible for measurement. Even for the largest earthquakes, these individual temperature pulses would be indistinguishable from background a few months after the event, and so timely attempts to detect them by drilling would be difficult (McKenzie and Brune, 1972; Lachenbruch, 1986). However, because the frictionally generated heat diffuses quite slowly, it should accumulate in the vicinity of the fault, eventually building up the local thermal gradient until the observable heat loss at the Earth's surface in the fault zone exceeds the normal background heat flow by the rate of heat generation on the fault. Thus, in principle, the measurement of a heat-flow anomaly in the fault zone should permit an estimate of the average frictional contribution τ_r to the earthquake fault stress (eq. 5).
      The heat-flow anomaly that we seek does not depend on the amount of heat E_r liberated by a single earthquake in a restricted fault area with a displacement u (eq. 4), but on the long-term average rate of heat generation (Q) and the long-term average slip rate (v) from the cumulative effect of successive events. Although most fault displacement probably occurs within a few tens of seconds during large earthquakes every century or so at slip velocities greater than the average ones by a factor of about 10⁸ , the long-term buildup of the heat-flow anomaly would be indistinguishable from that caused by uniform slip at an equivalent average velocity because the thermal time constant for the buildup (approx 10 6 yr) is large relative to the earthquake-recurrence interval (10¹-10³ yr). Therefore, we view the slip as a uniform continuous process and introduce

Differentiation of equation 4 yields

where v is the long-term average slip velocity, τ_r is the (displacement averaged) frictional resistance, and Q is the long-term average rate of frictional heat production per unit area on the fault surface.
      Equation 13 refers to the entire seismogenic (brittle) layer (approx 10-15 km thick), not just a patch as in equation 5. Over this depth, it is reasonable to consider the long-term slip velocity v to be independent of depth, but generally the heat-production rate Q will not be. For example, if the friction τ_r increased linearly with depth (for example, because of increasing gravitational pressure on the fault, as discussed below), the heat production Q on the fault would also increase linearly, as shown in figure 10.4B. According to heat-conduction theory, the temperature in the fault plane would then build up over time, as shown in figure 10.4C, and a heat-flow anomaly would develop at the surface over the fault, as shown in figure 10.4A. For such a distribution, a sharp heat-flow anomaly is seen to build up over the fault in about 1 m.y.; after several million years, it approaches a maximum value somewhat greater than half the average frictional heat production Q on the fault surface (fig. 10.4B). This anomaly falls off over a distance from the fault of the same order as the depth of the seismic layer (assumed to be 14 km in fig. l0.4). Other reasonable distributions of frictional sources give similar results (see Henyey, 1968; Brune and others, 1969; Henyey and Wasserburg, 1971; Lachenbruch and Sass, 1973, 1980).

Figure 10.4 - Surface heat flow q (A) and fault-plane temperature Θ (C) for a linear increase in source strength from zero at the surface to Q2 at depth x2 (B). t, time since initiation of faulting; λ, dimensionless time; Q, average rate of frictional heat generation on fault; K, thermal conductivity. Dimensional results are for x2=14 km, K=2.5 W/mK, and Q=42 mW/m2 (equivalent to 2v25 mm/yr, τr50 MPa) (Lachenbruch and Sass, 1980).

      The long-term slip rate v, which can be estimated from studies of offset (and dated) geologic features, generally ranges from 2 to 4 cm/yr for motion on the main trace of the San Andreas fault over the past several million years throughout California (see chap. 7; Grantz and Dickinson, 1968). As a useful rule of thumb, if the fault in figure l0.4 were slipping at an average rate (v) of 3 cm/yr and resisted by an average frictional stress τ_r of 100 MPa, then the average rate of frictional heat production Q (figure 10.4B) would be about 0.1 W/m²; that is,

This quantity is about twice the stable continental heat flow, and so, according to figure 10.4A, the corresponding heat-flow anomaly over the fault would be about 100 percent of background after 2 or 3 m.y. of fault slip, whereas if the mean frictional resistance were only 10 MPa, the corresponding heat-flow anomaly would be only about 10 percent of background, close to the limit of detection. Accordingly, if no heat-flow anomaly could be detected over the fault, the mean frictional resistance would be no more than about 10 MPa; if the mean frictional resistance were about 100 MPa, a very conspicuous anomaly should be observed.
      An example of heat-flow measurements near the San Andreas fault is shown in figure 10.5 for the Mojave Desert region of southern California (region 7, fig. 10.6). The pattern of anomaly curves from the model in figure l0.4 is scaled for the estimated local slip velocity (25 mm/yr; Weldon and Sieh, 1985) and for a mean frictional resistance of 50 MPa. Clearly, the data are incompatible with such an anomaly; in fact, the average heat flows near the fault and far from the fault ("within 10 km" and "beyond 10 km," figs. 10.7C, 10.7D) are statistically indistinguishable. Figures 10.7A and 10.7B show that the same condition prevails in the Coast Ranges of central California (regions 3-6, fig. 10.6). In fact, no local heat-flow anomaly has been confirmed anywhere on the main trace of the San Andreas fault (for possible exceptions, see Lee, 1983; Lachenbruch and Sass, 1988), and so, according to the foregoing simple considerations, the average friction on the fault, τ_r, probably does not exceed 10 MPa.

Figure 10.5 - Heat flow as a function of distance from the San Andreas fault in the Mojave segment (region 7, fig. 10.6). Theoretical anomaly is for a slip velocity of 25 mm/yr and average friction of 50 MPa (Lachenbruch and Sass, 1988).

Figure 10.6 - Locations of heat-flow measurements near the San Andreas fault and of numbered regions referred to in figure 10.7 (Lachenbruch and Sass, 1980). Heavy line, San Andreas fault, dashed where approximately located, dotted where concealed; stippled area, Great Valley.

Figure 10.7 - Comparison of heat flow within 10 km of main trace of the San Andreas fault (A, C) and beyond 10 km (B, D) for regions 3-6 (A, B) and region 7 (C, D) (see fig. 10.6 for locations); transitional regions 1 and 2 are not represented. Modified from Lachenbruch and Sass (1980). n, number of samples; , mean heat flow; SE, standard error.

      In summary, we note that analysis of the kinetic energy of seismic waves suggests that the associated apparent stresses (τ_a) do not exceed 5 MPa. Similarly, analysis of long-term frictional heating and the predicted and observed effects on heat flow from conduction theory suggest that the average frictional resistance τ_r does not exceed about 10 MPa. Thus, according to equation 5, the long-term average combined earthquake stress probably does not exceed about 15 MPa, and, of course, it could be much less. The initial stress τ₁ or "fault strength," would be greater by half the stress drop (fig. 10.2; Lachenbruch and Sass, 1980, eq. 41a), no more than another 5 MPa, for an upper limit of 20 MPa. The major assumptions in this analysis are (1) that heat transfer is exclusively by conduction-that is, no significant heat is transferred by moving ground water; (2) that the fault geometry can be represented by the usual simple conventions (see figs. 10.2, 10.3, and 10.4); and (3) that an earthquake's energy is converted exclusively to seismic waves and heat-that is, no appreciable energy is consumed by creating new surfaces, phases, or chemical combinations (Lachenbruch and Sass, 1973, 1980). We shall discuss these points later.

      We have seen that the average shear stress on an earthquake fault can be viewed as the sum of a dynamic part τ_a and a frictional part τ_r. The dynamic part is shown to be small from seismic evidence, and so the earthquake stress must be large or small according to the size of τ_r. We have also seen that τ_r is small according to geothermal evidence. We now consider a second line of evidence from laboratory measurement of friction which suggests to many that, contrary to the geothermal evidence, τ_r must be large.
      According to these results, rock surfaces will slide when the shear stress on their surface of contact exceeds the static frictional strength τ_f, given by

σ_n is the normal pressure pushing the surfaces together, and P is the fluid pressure in the pores and cracks tending to hold the surfaces apart; σ'_n is called the "effective" normal stress (we generally denote such effective stresses by a prime, " ' "). The proportionality constant μ in equation 15a is the coefficient of static friction; extensive laboratory experiments show that its value is generally in the range 0.6-0.9 for a remarkably large variety of rock types and surface conditions (Byerlee, 1978), although some studies (for example, Wang and others, 1980), reported substantially lower friction coefficients for some geologic materials, including certain types of fault gouge.
      We presume that a fault is a fracture with little cohesive strength that remains inactive until the natural shear stress τ resolved along it exceeds its frictional strength τ_f given by equation 15. This shear stress, which depends on the magnitudes of the principal stresses and on the angular relation between the fault plane and the principal stress directions (fig. 10.8), is given by (Jaeger, 1956, p. 8)

      where it is assumed for convenience that the intermediate-principal-stress direction (σ₂) lies in the fault plane (true if μ is independent of the orientation of this plane). In figure 10.8, σ₂ is vertical, and σ₁ and σ₃ are the maximum and minimum horizontal principal stresses. Θ is the angle formed by the fault normal and the direction of least compression (σ₃); it is also the angle between the fault trace and the direction of greatest compression (σ₁). To express the failure criterion (eq. 15) in terms of the stress field and fault orientation, we note that the effective normal stress, σ'_n, in equation 15 can be written as (Jaeger, 1956, p. 8)

With equations 15 through 17, the friction stress τ_f that must be exceeded on a fault for it to slip can be determined if we know (1) the maximum and minimum principal stresses σ₁ and σ₃, (2) the fluid pressure P, (3) the coefficient of friction μ, and (4) the angle Θ describing the orientation of the fault relative to the principal-stress axes.

Figure 10.8 - Conventions for discussing orientation of fault relative to direction of principal stresses: σ1>σ2>σ3. Arrows indicate direction of relative movement along fault.

      As we increase the stress difference, in what direction (Θ) will the Earth ultimately break, and what will be the stress on the failure plane? Clearly, the answer could be influenced by the existence of planes of weakness (Mc- Kenzie, 1969); for example, major preexisting faults or foliated country rock might result in directions with anomalously low μ.

      We first assume that no such directional strength variation exists, that the rock is fractured in all directions, and that all potential shear surfaces have the same coefficient of friction μ. In this case, the foregoing equations show that the trace of the favored fault plane will depart from the direction of maximum compression by an angle Θ₀, dependent only on the coefficient of friction, as follows:

Note that generally Θ₀<45° (the direction of the surface of maximum resolved elastic shear stress, eq. 16) because of the effects of normal stress on friction (Jaeger, 1956). With this additional relation (eq. 18), we can express the frictional strength τ_f of a plane of orientation Θ in terms of the coefficient of friction and the effective-principal-stress components as follows:

      To evaluate the frictional strength, the vertical stress is generally assumed to be a principal stress (reasonable because the Earth's surface supports no traction) equal to the rock column's weight, ρgz, an assumption supported by in place stress measurements (McGarr and Gay, 1978). In this case, the vertical effective stress σ'_v will be

where ρ is the rock density, and the fluid pressure P is given by

The value λ=0 represents conditions in dry rock. For a typical open ("hydrostatic") hydrologic system, we have λ∼0.37 (= ρ_w/ρ_rock, where ρ_w is the density of water). As λ→1, the fluid pressure approaches the weight of over-burden, and the vertical effective stress σ'_v vanishes (as discussed below, this limit probably occurs only in the thrusting regime, where (σ₃ is vertical).
      The curves in figure 10.9 (referred to ordinate scale at left margin) give the frictional strength normalized by the effective vertical principal stress for those cases in which the vertical stress is the maximum (dashed curve), average (solid curve), or minimum (dotted curve) principal stress, respectively. The first right-hand ordinate scale gives the increase in frictional strength with depth (τ_f/z) for the usual assumption of hydrostatic fluid pressure (P=ρ_wgz). For typical values of μ from Byerlee's results (for example, 0.6-0.9), the frictional strength for normal and thrust faults increases with depth at rates of about 5 and 20 MPa/km, respectively (fig. 10.9). The rate of increase for strike-slip faults lies between these limits; a commonly used value, 8 MPa/km, is shown by the solid curve in figure 10.9. For an upper-crustal fault extending to 14 km depth, these increases would result in average friction (the value at a 7-km depth) of 35, 56, and 140 MPa for normal, strike-slip, and thrust faults, respectively (see second ordinate scale on right, fig. 10.9). Such calculations provide the basis for the expectation of high fault stress from the analysis of laboratory results: These values are substantially greater than the 20 MPa upper limit for initial stress suggested from the analysis of heat-flow data in strike-slip tectonic regimes (horizontal dashed line, fig. 10.9). Note that the heat-flow limit would require μ0.2 for the assumed conditions.

Figure 10.9 - Variation of normalized fault friction (left-hand ordinate scale) with coefficient of friction under three conditions for vertical effective stress σ'v) for failure at optimum angle Θ0, Increase in fault friction τf with depth z under hydrostatic fluid pressure is given by right-hand ordinate scale, and mean friction f on a fault 14 km deep by scale on far right. Horizontal dashed line is upper limit of mean friction suggested by heat-flow constraint. σ'1 and σ'3 maximum and minimum horizontal effective principal stress, respectively.

      The estimates of large friction from the fault model of figure 10.9 depend on three principal assumptions: (1) that the average coefficient of friction on real faults is comparable to typical laboratory values (μ∼0.6-0.9), (2) that the average fluid pressure throughout the depth of the fault is comparable to the weight of the overlying column of water (λ∼0.37, eq. 20), and (3) that the coefficient of friction (μ) is the same in all directions, so that the fault direction (Θ₀) is determined by the applied stress (eq. 18) and not by the orientation of a special plane of weakness. Partly in response to recent reports that the maximum horizontal principal stress is oriented nearly perpendicular to the San Andreas fault (Mount and Suppe, 1987; Zoback and others, 1987), we drop the last assumption and suppose that the fault occupies a very weak plane (which is assumed to contain the intermediate principal stress). Because of the anomalous weakness of this plane, the friction along it could be very low, consistent with the heat-flow data, and faulting could persist there irrespective of the ambient stress field. According to the friction model (eq. 15), the two factors that might weaken the plane are either an abnormally low coefficient of friction or unusually high pore pressure. For now, we assume that each of these conditions can exist regardless of laboratory or hydrologic evidence.
      The first question we consider is whether a very weak fault can coexist with stronger faults such that both types are active, as may be the case along the San Andreas fault (for a closely related discussion, see Sibson, 1985). To address this question, it is convenient to express the crustal strength in terms of the ratio σ'₁/σ'₃ (Brace and Kohlstedt, 1980). From equations 19a and 19b, the condition at failure (eq. 15a) for a weak plane oriented at an angle Θ to the direction of σ₁ (fig. 10.8) is

For isotropic strength, failure occurs at Θ₀ (eq. 18), the direction in which σ'₁/σ'₃ is a minimum for a given μ:

Faults at angles other than Θ₀ support greater deviatoric stresses and the higher values of σ'₁/σ'₃ given by equation 21a.
      The conditions necessary for the coexistence of active faults with different coefficients of friction are illustrated in figure 10.10, where the ratio of effective principal stresses at the point of failure is plotted as a function of the fault angle for various values of the coefficient of friction (eq. 21a). Suppose, for example, that the coefficient of friction is only 0.1 in the direction of the San Andreas fault, whereas in all other directions it is 0.6. Because σ'₁/σ'₃ must be at least 3.1 to cause faulting in the crustal environs, the low-strength San Andreas fault must be oriented at Θ≤3.5° or Θ≥81.5° (fig. 10.10); otherwise, σ'₁/σ'₃ would be too low to cause slip in the stronger directions. In this example, then, the weak fault must be oriented either nearly parallel or nearly perpendicular to the direction of σ₁.

Figure 10.10 - Failure criteria for various coefficients of friction as functions of fault angle Θ. For fixed σ'1, σ'3, and μ, slip will occur for all Θ between points where horizontal line defined by σ'1/σ'3 intersects curve defined by μ. For example shown, if σ'1/σ'3=3.1, then for μ=0.1, slip can occur at all angles between 3° and 82° but only at optimum angle 29.5° if μ=0.6. Dashed curve, optimal failure angle Θ0(μ) (eq. 18). σ'1and σ'3, maximum and minimum horizontal effective principal stress, respectively.

      In the context of the notion that the San Andreas fault is nearly perpendicular to the direction of σ'₁, or at Θ∼90° in figure 10.10, we note that a very low coefficient of fault friction is required. The strength curves for each value of μ have two asymptotes where σ'₁/σ'₃ → ∞. These asymptotes occur where the denominator of equation 21a vanishes; one asymptote is at Θ=0, or σ₁ parallel to the fault, for any value of μ, and the other is at Θ=2Θ₀ (eq. 8), or Θ=90-tan^-1 μ. Thus, the normal to any fault that fails in shear must be oriented at an angle of at least tan_-1 μ from the direction of σ₁. For the four curves in figure 10.10, the right-hand asymptotes are at Θ=84.3, 73.3, 59.0, and 48.0, respectively, for μ=0.1, 0.3, 0.6, and 0.9. Thus, if the fault trace makes an angle greater than 59°, then μ must be less than 0.6 as long as the fluid pressure is less than the least principal stress.
      More generally, enhanced pore pressure alone cannot lead to active faults nearly normal to σ₁ unless P>σ₃, in which case σ'₃<0 and failure is likely to manifest itself as hydraulic fracturing rather than fault slip.

      To recapitulate, the simplest interpretation of earthquakes in terms of the frictional fault model and laboratory measurements of rock friction leads to fault stresses many times larger than the limits suggested from heat-flow and fault energetics. This interpretation depends on three assumptions: (1) that the average coefficient of friction on real faults is comparable to typical laboratory values (μ∼0.6-0.9), (2) that the pore-fluid pressure throughout the depth of faulting is comparable to the weight of the overlying column of water, and (3) that the intrinsic resistance of the Earth to sliding is isotropic- that is, no weak directions exist. To reduce the high estimates of friction obtained from rock mechanics to the low ones obtained from heat flow, we must assume either smaller values of the coefficient of friction μ or larger values of fluid pressure P (see eq. 15). Of particular interest in this connection is reported evidence that the trend of the San Andreas fault in California might occupy an anomalous weak direction. According to Mount and Suppe (1987) and Zoback and others (1987), the fault plane is nearly perpendicular to the direction of maximum compression (Θ∼π/2, fig. 10.10), a direction in which the resolved shear stress is very small. Such a condition could be consistent with the low friction required by heat flow, while permitting high stresses to accommodate the subsidiary faulting observed on more favorably situated planes. This model, however, raises some basic questions regarding the mechanics of faulting; for the fault to slip, it must have a low shear strength, as well as the low shear stress suggested by its orientation. If conventional friction theory applies, anomalously high fluid pressure along the fault cannot readily account for the required low friction because, unless μ is unusually low, the fault becomes exceedingly resistant to shear failure as Θ begins to approach π/2 (fig. 10.10).

      In principle, measurements of the magnitude and orientation of crustal stress in the vicinity of the San Andreas fault should provide the most direct evidence of the forces acting to cause interplate motion there. However, some essential problems exist with this approach. Because we have little understanding of the mechanics of the system, it is difficult to interpret the data. We are not dealing with a laboratory experiment in which a sample is loaded in a testing machine whose characteristics are well known; in such a situation, it is straightforward to use gages to estimate the magnitude of the load. In contrast to the well-controlled laboratory situation, we have little idea of the nature of the forces applied to the Earth's crust to cause a deviatoric state of stress and, in the case of tectonically active areas, slip across major throughgoing faults. We know neither where the forces are applied nor what is applying them; moreover, there is even debate about what the state of stress would be if only gravity were acting (McGarr, 1988).
      In addition to the absence of a conceptual framework, there are numerous experimental difficulties in determining the state of stress, that is, the magnitudes and orientations of the three principal stresses as functions of position within the crust. Data must be obtained from depths below the zone of weathering, in rock that is sufficiently strong to support deviatoric stresses. In granitic rocks, this requirement, in effect, necessitates stress measurements at depths of about 50 m or more, thus limiting the measurement technique to hydraulic fracturing, the only common procedure that can be used at such depths (Haimson and Fairhurst, 1970).
      The hydraulic-fracturing, or "hydrofrac," method involves isolating a section of a borehole and then pressurizing this cylinder by pumping in fluid until a tensile crack forms and propagates into the previously unfractured rock. By monitoring the pressure-time history of the fluid in the isolated section, both the maximum and minimum horizontal stresses can be estimated (Hubbert and Willis, 1957; Zoback and Haimson, 1983). This approach assumes that one of the principal stresses &sigma_v is oriented vertically and can be calculated from the weight of overburden (eq. 20). The other two principal stresses are the maximum, σ_H, and minimum, σ_h, horizontal stresses. In contrast to engineering usage, the convention adopted here is for compressional stresses to be positive because, in the Earth's crust, tensional stresses are rarely encountered, even at the surface.
      Although the uppermost crust near the San Andreas fault system has not been sampled as much for stress as for heat flow, enough inplace stresses have been measured to provide an indication of the state of stress there and how it compares with crustal stresses in other tectonic settings. To date, 41 successful hydrofrac measurements have been made in the 12 wells shown in figure 10.11 at depths of as much as 850 m. A total of 29 of these data, in wells along the Mojave reach of the fault (fig. 10.11C), were analyzed by McGarr and others (1982). Since that study, four stress measurements have been made at Black Butte (BB, fig. 10.11C) in the Mojave Desert (Stock and Healy, 1988), the data from the Hi Vista well have been reanalyzed by Hickman and others (1988), and additional measurements have been made in central California (Zoback and others, 1980). Currently, stress measurements are being made at the Cajon Pass well near the southeast end of the Mojave reach of the San Andreas fault, with some observations at depths below 3 km. Because no clear picture has yet emerged (see Healy and Zoback, 1988), we have not incorporated the Cajon Pass results into this review.

Figure 10.11 - Sketch maps of the Gabilan Range (A), central California coast (B), and western Mojave Desert (C), showing locations of wells where stress measurements have been taken along the San Andreas fault using the hydrofracturing technique.

      The state of horizontal deviatoric stress can be characterized in terms of two parameters: the maximum horizontal shear stress τ_m given by

and the angle Θ between the trace of the fault and the direction of maximum horizontal compressive stress σ_H. Under favorable conditions, both parameters can be determined by the hydrofrac technique. We have shown that if Θ∼45°, then τ_m is entirely resolved onto the plane of the fault to produce its slip; as Θ approaches 0° or 90°, the resolved stress on the fault becomes arbitrarily small irrespective of the magnitude of τ_m (eq. 16).
      Evidence regarding the actual orientation of σ_H relative to the strike of the San Andreas fault is contradictory. Observations favoring Θ distributed about 45°, so as to cause dextral fault slip, were presented by McNalley and others (1978), Zoback and others (1980), Zoback and Zoback (1980), and Hickman and others (1988); however, these data, from the Mojave Desert, show considerable scatter. In contrast, Mount and Suppe (1987) and Zoback and others (1987) reviewed a broad set of data, including many borehole breakout orientations, that suggest Θ=90°; Oppenheimer and others (1988) came to a similar conclusion. An intermediate result was obtained by Jones (1988), who stated that σ_H is oriented at 65° to the local strike of the San Andreas fault in southern California. Thus, currently, we know neither the preferred value of Θ nor whether such a value even exists. For the foregoing values of Θ (45°, 65°, or 90°), the shear stress resolved on the fault (eq. 16) would be τ_m, 0.77τ_m, or 0, respectively. In view of this uncertainty, we leave Θ unspecified and describe what is known of τ_m the upper limit to the shear stress that can be resolved on the fault.
      The first-order feature seen in data from the San Andreas fault zone (fig. 10.12) is a marked tendency for τ_m to increase with depth. The solid line, a regression fit to all of the data, indicates a depth gradient of 8.3 MPa/km, not significantly greater than the gradient of 7.9 MPa/km reported by McGarr and others (1982) on the basis of 29 of the 41 data plotted in figure 10.12. We note that the observed depth gradient of τ_m also agrees well with the curves for strike-slip faults (solid curves, fig. 10.9) for a coefficient of friction of 0.6 or greater. In addition to the general increase in τ_m with depth, considerable variation from one well to another and within individual wells is suggested by figure 10.12.
      Figure 10.13 shows that the departure of the measured values of τ_m from the regression line in figure 10.12 does not vary systematically with distance from the San Andreas fault. The principal conclusion to be drawn from figure 10.13 seems to be that the magnitude of deviatoric stress is not measurably affected by proximity to the San Andreas fault. Thus, whatever effect the fault may have on the magnitude of the shear stress, it is either too subtle, too localized, or too deep to be recognized in the current data set.

Figure 10.12 - Maximum horizontal shear stress as a function of depth z. Solid regression line has been fitted to all data, and dashed line to measurements in crystalline rock only. See figure 10.11 for locations of wells.

Figure 10.13 - Averages of N maximum-shear-stress (τm) residuals as functions of distance from the San Andreas fault. In calculating residual at a particular distance, effect of depth z is removed by using solid-line regression fit to data of figure 10.12 [τm (MPa) = 1.15+8.28z (km)]. Error bars, 1σ.

      We note that there is no detectable difference between the Mojave Desert residuals, measured near a locked section of the San Andreas fault, and those in central California (fig. 10.11A), where the fault is creeping and presumably does not produce great earthquakes. If measurements were made to greater depths, some differences might appear, but at least in the topmost several hundred meters, the magnitude of shear stress seems to be largely independent of position along the strike of the San Andreas fault.
      Having failed to discover any spatial relation between the San Andreas fault and deviatoric-stress magnitudes, we now consider the question of whether or not any detectable differences exist between the stress states measured near the San Andreas fault (fig. 10.12) and those measured elsewhere in different tectonic settings. A review of crustal shear stress by McGarr (1980) considered a large suite of stress data in "hard" rocks measured at depths extending to 3.6 km. The resulting regression line of

has a greater surface intercept but a similar depth gradient to the San Andreas regression

fitted to the crystalline-rock data in figure 10.12. The comparison between equations 23 and 24 is not entirely appropriate because the data used to develop equation 23 represent all three stress states; stresses measured in regions of strike-slip tectonics were not considered separately by McGarr (1980) (see fig. 10.9). More recently, however, data measured in a 2,000-m-deep well in Cornwall, U.K. (Pine and others, 1983), permit quite an interesting comparison. For both the San Andreas and the Cornwall data sets, most of the stress observations are compatible with strike-slip tectonics; that is, σ_v is the intermediate principal stress. For most of the San Andreas and all of the Cornwall measurements, the rock is granitic. In contrast to the San Andreas system, however, the tectonic setting in Cornwall is presently inactive. The deviatoric stresses at Cornwall are believed to be a consequence of the Alpine orogeny, which apparently has caused the maximum horizontal stress to be oriented northwestward throughout much of Europe (for example, McGarr and Gay, 1978).
      The 12 data sets obtained by Pine and others (1983) indicate a stress state (fig. 10.14) surprisingly similar to that of the San Andreas fault (fig. 10.12). For the maximum shear stress, the depth gradient of 7.52 MPa/km is indistinguishable from its counterpart in the crystalline San Andreas crust of 7.46 MPa/km; however, the surface intercept at Cornwall is larger. If the state of deviatoric stress is much the same in Cornwall as along the San Andreas system, then we must conclude that the plate-tectonic motion in California along the San Andreas fault has no expression in the shallow (1-2 km deep) stress field. Accordingly, much of what has been discovered about continental-crustal stress in general may apply to the crust adjacent to the San Andreas fault.

Figure 10.14 - Maximum horizontal shear stress τm as a function of depth z in granite near Cornwall, U.K. Data from Pine and others (1983).

      This generalization implies that the applied forces which give rise to τ_m in the vicinity of the San Andreas fault are not specific to the Pacific-North American plate boundary. In terms of observed shear stress, a major active plate-boundary fault would be at least as likely in Cornwall, U.K., as in California from what we currently know of stress magnitudes, at depths down to a few kilometers.

      From what we currently know of crustal stress and heat flow, neither is influenced by proximity to the San Andreas fault, the most conspicuous and best studied plate-boundary fault on the continents. The measured horizontal shear stress increases rapidly with depth (approx 8 MPa/km), essentially as would be predicted from laboratory measurements of friction and the assumption that crustal stress is limited by the frictional resistance of fractures forced together by the weight of overlying rocks. From this consistency of independent observations, two large "ifs" lead to what seems to be a physical contradiction: (1) if these vertical stress gradients persist throughout the depth of the seismogenic faulting layer (approx 12-15 km), then the average of the maximum horizontal shear stresses throughout the layer is quite large (approx 50 MPa); and (2) if the direction of the San Andreas fault is aligned with this maximum-horizontal-shear-stress direction, then the frictional heat generated by such stress during the documented fault motion (tens of kilometers per million years) should cause the background heat flow to double as the fault is approached. In 100 heat-flow measurements over a 1,000-km span of the San Andreas fault, no such heat-flow anomaly has been observed.
      The contradiction stems from two separate lines of argument: (1) in place and laboratory measurements of rock stress imply average fault stresses of about 50 MPa or more, and (2) the absence of a local heat-flow anomaly and the energy balance of the fault imply an average fault stress of about 15 MPa or less. At least one of these arguments must be wrong. We have outlined the major factors in each argument, and we shall now point out some possible loopholes and areas for further study.
      The energy-balance argument leading to the heat-flow constraint on fault stress could be invalidated if the neglected energy sinks turn out to be important, or if the heat-conduction model is unrealistic or inappropriate. The general energy argument assumes that fault slip produces only seismic radiation and heat. It supposes that the energy consumed by the grinding of rocks into fault gouge (Lachenbruch and Sass, 1980) or the heat absorbed by possible phase changes or chemical reactions is negligible, and that the energy of seismic radiation does not grossly differ from the estimates made by seismologists. The heat-conduction model assumes that the frictional heat production occurs in a near-vertical fault zone (whose width is small relative to its depth) extending throughout the seismogenic layer. Systematic nonconductive removal of frictional heat by circulating ground water could invalidate this model (see Q'Neill and Hanks, 1980; Williams and Narasimhan, 1989), as could a grossly different fault geometry-for example, a fault whose lower half was continually being rejuvenated because of migration of the upper half away from it along an upper-crustal detachment surface (Namson and Davis, 1988). All of these effects probably deserve further study.
      The mechanical argument leading to large fault stress is based on observations of inplace stress (to maximum depths of approx 1 km in the San Andreas fault zone and of approx 4 km elsewhere on the continents), on laboratory measurements of rock friction and the efficiency of simulated earthquakes, and on downward extrapolation of these results through the seismic layer, on the assumption that fluid pressure is normal and frictional properties are uniform and isotropic. The consistency between the most frequently measured friction coefficients and the inplace determinations of the vertical gradient of maximum shear stress is reassuring (solid curves, fig. 10.9; fig. 10.12); however, the downward extrapolation of these results to depths of 10 or 15 km is an uncertain step, with loopholes that could invalidate the high-fault-stress conclusion.
      There are at least three such loopholes. First, the fluid pressure might increase with depth, as it is known to do in some sedimentary basins, approaching the minimum principal stress (Berry, 1973). Second, the friction coefficient at depth might be lower than average laboratory values; such lower values have been reported in some studies of gouge and other clay-size aggregates (Wang and others, 1980). Each of these effects could substantially lower the maximum stress at depth. Third, the frictional strength properties might be anisotropic, with the main trace of the fault occupying a weak direction. If so, the maximum stress at depth might be high, as maintained in the mechanical argument, but the shear stress resolved on the fault might be very low, as maintained in the thermal argument. Under these circumstances, the fault must be nearly parallel to a principal axis, as suggested by Mount and Suppe (1987) and Zoback and others (1987). Such a condition could be consistent with the low friction on the main fault required by heat flow, while permitting high stresses to accommodate subsidiary faulting on more favorably situated planes with normal frictional properties.
      As mentioned above in the section entitled "Introduction," geothermal studies of the San Andreas fault have provided evidence for a very weak fault for two decades; the meaning of this result depends heavily on the direction of principal stresses in the fault zone and the magnitude of the stress differences there. As we have shown, existing evidence is contradictory, especially in the Mojave Desert region. Many measurements of stress near the San Andreas fault suggest that the fault trace is inclined at an intermediate azimuth (approx 45°) to the principal-stress directions, and thereby imply that the fault coincides approximately with the direction of maximum shear stress and that the heat-flow constraint could not be satisfied unless horizontal shear stress (and stress differences) were low everywhere. In this case, both the San Andreas fault and active subsidiary faults with other orientations would have to be weak. If, however, the horizontal-stress differences are large, the weak fault required by heat flow is a "zero shear stress" boundary condition on the adjacent fault blocks that requires the fault to be almost normal to a principal-stress direction. In this case, the heat-flow constraint could be honored on an anomalously weak main trace, whereas subsidiary faults with other orientations and normal strength could also be active.
      Thus, the occurrence of a weak direction may reconcile observations of rock mechanics with the longstanding implications of thermomechanical studies. It does, however, raise several questions:

REFERENCES CITED