Frictional Equilibrium: The Crust at Its Limit

Part 5, Part 5: In-Situ Stress, The Polygon

Learning objectives

Show that friction on well-oriented faults caps the ratio of the effective principal stresses
Derive the limit q, the maximum effective stress ratio a friction coefficient can sustain
Evaluate the crustal limit: mu of 0.6 gives q of 3.12, mu of 1.0 gives 5.83
Explain why overpressure weakens the crust and hydrostatic pressure does not change the limit

Friction Caps the Spread

Anderson's regimes name the ordering of the stresses, but not how far apart they can spread. Byerlee's law, from Part 3, supplies the missing bound. The crust is threaded with faults of every orientation, and a fault optimally oriented for slip will fail once the shear on it reaches friction times the effective normal stress. Push the stress difference higher and that fault slips, relieving it. So the crust cannot sustain an arbitrary spread between its largest and smallest effective stresses; friction caps the ratio at the value where the best-oriented fault is on the verge of sliding: $\dfrac{\sigma_1'}{\sigma_3'} \le \left(\sqrt{\mu^2 + 1} + \mu\right)^2 = q$ leleft(sqrtmu2+1+muright)2=q. This is the same $q$ that set the slope of the Coulomb line; here it is the crust's frictional speed limit, the maximum any principal stress can exceed any other once pore pressure is subtracted.

Turn the friction dial in the figure. At the crustal $\mu = 0.6$ , the limit ratio is 3.12: the largest effective stress can be at most 3.12 times the smallest before a well-oriented fault slips. Raise the friction to $\mu = 1.0$ and the limit opens to 5.83; drop it toward the clay-gouge values and it closes toward 2. The Mohr inset shows the geometry directly: at the limit, the stress circle is exactly tangent to the friction line, the same kiss-the-envelope condition as intact failure but now for sliding on a pre-existing surface. A crust held at this limit is called critically stressed, and the remarkable observation, from stress measurements in deep wells worldwide, is that much of the crust is critically stressed: it has faulted until friction alone holds it, so the measured stress states press right against the 3.12 limit.

Why Pressure Is the Lever

The limit is written in effective stresses, and that placement is the whole story of induced seismicity. Because $q$ bounds the ratio of the primed stresses, adding pore pressure does not just shrink both effective stresses equally; it shrinks the denominator $\sigma_3'$ 3′ faster in relative terms, driving the ratio up toward the limit. A crust sitting comfortably below $q$ at hydrostatic pressure can be pushed to it by overpressure or by injection, which is exactly how wastewater disposal wakes faults, as Part 9 will compute. Conversely, at purely hydrostatic conditions the whole ladder scales together and the limit is not approached by pressure alone. The lesson the polygon of the next section will make visual: the crust's allowable stress states are bounded by friction through the effective stresses, and pore pressure is the lever that moves a state toward or away from that bound. With the ordering from Anderson and the spread limit from friction, we can now draw every state the crust permits.

References

Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
Byerlee, J. (1978). Friction of rocks. Pure and Applied Geophysics, 116(4-5), 615-626.
Townend, J., & Zoback, M. D. (2000). How faulting keeps the crust strong. Geology, 28(5), 399-402.