THE CASE OF EQUAL STRENGTH IN ALL DIRECTIONS

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 mu. 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 theta0, dependent only on the coefficient of friction, as follows:

(18)

Note that generally theta0<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 tauf of a plane of orientation theta in terms of the coefficient of friction and the effective-principal stress components as follows:

(19a)

(19b)

(19c)

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, pgs, an assumption supported by inplace stress measurements (McGarr and Gay, 1978). In this case, the vertical effective stress sigmav' will be

(20a)

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

(20b)

The value lambda = 0 represents conditions in dry rock. For a typical open ("hydrostatic") hydrologic system, we have lambda ~ 0.37 (= pw prock, where pw is the density of water). As lambda ~ 1, the fluid pressure approaches the weight of over-burden, and the vertical effective stress sigmav vanishes (as discussed below, this limit probably occurs only in the thrvsting regime, where sigma3 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 (tauf/z) for the usual assumption of hydrostatic fluid pressure (P= pwgz). For typical values of mu 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 mu <= 0.2 for the assumed conditions.