Effective Stress Moves the Circle

Part 1, Part 1: Stress, the Tensor and the Circle

Learning objectives

State Terzaghi's principle: the frame feels total stress minus pore pressure, on every normal component at once
Show in the Mohr diagram that pore pressure slides the circle left without changing its radius
Compute the canon effective stresses, 32.4 and 10.7 MPa, from the totals and the 35.3 MPa pore pressure
Explain why raising pore pressure moves a rock toward failure even though nothing pushed harder

What the Frame Feels

The Rock Physics course reached this idea from the velocity side in its Part 8.1, on exactly the ladder we share: overburden 67.7, hydrostatic pore pressure 30.3, and the difference, 37.4 MPa, is what the grain frame carries. Here is the same principle stated for the whole tensor. Pore fluid pushes outward on every grain equally, in every direction, so it relieves every normal stress by the same amount while leaving every shear stress untouched, because a fluid cannot drag. Terzaghi's effective stress is that subtraction: $\sigma' = \sigma - P_p$ p on the normals, $\tau' = \tau$ on the shears. It is the single most consequential equation in this book. The rock's deformation, its strength, its failure, all answer to the primed quantities.

Watch what that does geometrically. Both principal stresses drop by $P_p$ p, so the Mohr circle's center slides left by $P_p$ p while its radius, set by the difference of the principals, does not move a hair. With the canon totals, $S_v = 67.7$ v=67.7 and $S_{hmin} = 46$ hmin=46, and the canon pore pressure of 35.3 MPa, the center glides from 56.9 down to 21.6 MPa and the principal pair lands at $\sigma_1' = 32.4$ 1′=32.4 and $\sigma_3' = 10.7$ 3′=10.7 MPa, with the radius pinned at 10.85 throughout. Drag the pressure slider through its range and you are watching most of petroleum geomechanics: injection slides the circle left, depletion slides it right, and the shear it carries goes along unchanged.

Why Pressure Is Dangerous

Now put the two pictures of this part together. Strength, when Part 3 draws it, will be an envelope rising from the left side of the diagram: rocks are weakest where the normal stress that clamps their flaws and faults is smallest. A circle sliding left is therefore a circle sliding toward its envelope, at constant size. Nothing pushed harder; no load was added anywhere. The fluid simply unclamped the rock. That is how injection wakes faults in Part 9, how overpressure narrows the mud window in Part 6, and why the pore-pressure prediction of Part 4 is a safety discipline and not an accounting exercise. One refinement waits in Part 2: for the rock's volumetric deformation the subtraction is imperfect, and Biot's coefficient will meter it. For strength and friction, Terzaghi's full subtraction is the right tool and the course uses it.

References

Terzaghi, K. (1943). Theoretical Soil Mechanics. Wiley.
Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.
Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.