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285 Stress fields

original data compiled by Hubbert and Willis and equation (9.1) because the original
data available to them were from relatively shallow depths where pore pressures are
hydrostatic. The same thing is true for the data cited in west Texas by Eaton (1969). At
greater depth, however, where overpressure is observed in the Gulf of Mexico, frictional
faulting theory tends to underpredict measured values of the least principal stress, as
mentioned previously. This can be seen by comparing equation (9.6) with the K i values
used in equation (9.2)tofit data from great depth. As K i values get as high as 0.8–0.9
at great depth, it is clear that the theoretical value of the least principal stress predicted
by equation (9.6)is less than the data indicate.
As previously mentioned, one possible explanation for this is that if the coefficient
of friction of faults in smectite-rich shales is lower than 0.6, higher stress values would
be predicted using equation (9.6) and hence better fit the observed data at depth. For
example, a coefficient of friction as low as 0.2 results in an empirical coefficient in
equation (9.6)of 0.67, closer to the values for K i that should be used at depth as argued
by Matthews and Kelly (1967).
Finally, Holbrook, Maggiori et al.(1993) proposed a porosity based technique for
estimation of the least principal stress based on a force-balance concept:

S hmin = (1 − φ)(S v − P p ) + P p (9.7)
As porosity of overpressured shales is typically ∼35%, it yields similar values to that
predicted with K i ∼ 0.65 in the Matthews and Kelly (1967) relation for overpressured
shales at depth, but would seriously overestimate the least principal stress in the cases
presented in Figures 9.4–9.7.
Figure 9.9 presents calculated values (in units of equivalent mud weight in ppg) for
the magnitude of the least principal stress for an offshore well in the Gulf of Mexico
using the the formulae presented in Table 9.1 and discussed above. Input data include
the vertical stress (calculated from integration of the density log), pore pressure and
Poisson’s ratio (determined from P- and S-wave sonic velocity measurements). Curve a
illustrates the technique of Zoback and Healy (1984), curve b that of Breckels and Van
Eekelen (1981), curve c that of Hubbert and Willis (1957) using the modified empirical
coefficient of 0.5, curve d is that of Holbrook, Maggiori et al.(1993) and curve e is that
of Eaton (1969). At a depth of 4000 feet, where pore pressure is hydrostatic, there is a
marked variation between the predictions of the various techniques with the method of
Zoback and Healy (1984) (or that of Hubbert and Willis 1957 with an empirical constant
of 0.3) which yields the lowest values of S hmin (slightly in excess of 11 ppg). Recall from
the discussion above that where pore pressures are hydrostatic, the lower estimates of
the least principal stress seem to be more representative of measured values. Also, as
illustrated above in Figures 9.4–9.6, this technique seems to predict the least principal
stress well in cases of cemented rocks in normal faulting environments. Note also that at
this depth, the technique of Eaton (1969) predicts much higher values (about 14.5 ppg).
However, where pore pressure is elevated, these techniques consistently underestimate