THE CASE OF A WEAK DIRECTION

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 (mu ~ 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 (lambda ~ 0.37, eq. 20), and (3) that the coefficient of friction (mu) is the same in all directions, so that the fault direction (theta0) 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 crystal strength in terms of the ratio sigma1'/sigma3' (Brace and Kohlstedt, 1980). From equations 19a and 19b, the condition at failure (eq. 15a) for a weak plane oriented at an angle theta to the direction of sigma1 (fig. 10.8) is

(21a)

For isotropic strength, failure occurs at theta0 (eq. 18), the direction in which sigma1'/sigma3' is a minimum for a given mu:

(21b) Faults at angles other than theta0 support greater deviatoric stresses and the higher values of sigma1'/sigma3' 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 sigma1'/sigma3' must be at least 3.1 to cause faulting in the crustal environs, the low-strength San Andreas fault must be oriented at theta <= 3.5° or theta >=81.5° (fig. 10.10); otherwise, sigma1'/sigma3' 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 sigma1.

In the context of the notion that the San Andreas fault is nearly perpendicular to the direction of sigma1', or at theta ~ 90° in figure 10.10; we note that a very low coefficient of fault friction is required. The strength curves for each value of mu have two asymptotes where sigma1'/sigma3' --> infinity. These asymptotes occur where the denominator of equation 21a vanishes; one asymptote is at theta = 0, or sigma1 parallel to the fault, for any value of mu and the other is at theta = 200 (eq. 8), or theta = 90-tan-1 mu. Thus, the normal to any fault that fails in shear must be oriented at an angle of at least tan-1

mu from the direction of sigma1. For the four curves in figure 10.10, the right-hand asymptotes are at theta = 84.3, 73.3, 59.0, and 48.0, respectively, for mu = 0.1, 0.3, 0.6, and 0.9. Thus, if the fault trace makes an angle greater than 59°, then mu 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 sigma1 unless P > sigma3, in which case sigma3<0 and failure is likely to manifest itself as hydraulic fracturing rather than fault slip.