The Stress Polygon

Every State the Crust Allows, on One Diagram

The last two sections give everything needed for the single most useful diagram in in-situ stress. Anderson fixes the vertical stress $S\_v$; friction caps how far the horizontals can spread from it and from each other. Plot the two horizontal stresses against each other, $S\_{hmin}$ across and $S\_{Hmax}$ up, and the allowable states fill a bounded region: the stress polygon. Its edges are not arbitrary. The diagonal $S\_{Hmax} = S\_{hmin}$ bounds it above-left (the horizontals cannot cross). The point where both equal $S\_v$ anchors its center. And the frictional limit $q$ cuts the outer corners: in normal faulting $S\_{hmin}$ cannot fall below $(S\_v - P\_p)/q + P\_p$, and in reverse faulting $S\_{Hmax}$ cannot rise above $q(S\_v - P\_p) + P\_p$. Every stress state the crust can hold at this depth lives inside that outline; nothing outside it is physically sustainable.

This is the working canvas of the whole second half of the course, so learn to read it. The polygon divides into three regime triangles: normal faulting in the lower-left (both horizontals below $S\_v$), strike-slip in the middle band ($S\_v$ between them), reverse faulting in the upper-right (both above $S\_v$). At our canon $S\_v = 67.7$ and hydrostatic $P\_p = 30.3$ with $\\mu = 0.6$, the corners read a normal-faulting floor of $S\_{hmin} = 42.3$ MPa and a reverse-faulting ceiling of $S\_{Hmax} = 147.0$ MPa. Drop the canon reservoir state, $S\_{hmin} = 46$, $S\_{Hmax} = 62$, onto the diagram and it lands in the normal-faulting triangle, comfortably inside but not central, a real basin sitting toward the frictional edge.

Pore Pressure Breathes the Polygon

Now move the pore-pressure slider and watch the polygon breathe. Raise $P\_p$ and the frictional corners pull inward: the allowable region shrinks, because higher pore pressure means lower effective stresses and the friction limit bites sooner. Push $P\_p$ all the way to $S\_v$ and the polygon collapses to a single point at $(S\_v, S\_v)$, the crust unable to hold any stress difference at all when the fluid carries the entire load. That shrinking is the geometry behind every fault-reactivation and lost-well story in the course: a state that sat safely inside the polygon at hydrostatic pressure can find itself pressed against, or outside, the shrunken boundary when injection or overpressure raises $P\_p$. The polygon is not a static map; it is a live constraint that pore pressure reshapes, and the next two sections use it to do real work, bounding the one horizontal stress we can measure and the one we cannot.

References

• Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
• Moos, D., & Zoback, M. D. (1990). Utilization of observations of well bore failure to constrain the orientation and magnitude of crustal stresses. Journal of Geophysical Research, 95(B6), 9305-9325.
• Zoback, M. D., et al. (2003). Determination of stress orientation and magnitude in deep wells. International Journal of Rock Mechanics and Mining Sciences, 40(7-8), 1049-1076.