Stress and Anisotropy

When the Stress Is Not the Same in Every Direction

So far pressure has been a single number, the effective pressure squeezing the frame equally from all sides. Underground it rarely is. The vertical stress from the overburden and the two horizontal stresses are generally different, and the two horizontal stresses can differ from each other, especially near faults, salt, or a tectonic front. When the horizontal stress has a strong and a weak direction, the rock stops being the same in every direction: its velocity depends on the azimuth along which the wave travels. This is stress-induced anisotropy, and unlike the layer-induced kind of Part 9, it can change during production as the stress field shifts.

Cracks Do the Bookkeeping

The mechanism runs through the same soft microcracks that made velocity climb with pressure. A crack is compliant across its thin dimension and stiff along its face. Apply a maximum horizontal stress in one direction and it preferentially closes the cracks that face that direction, the ones whose thin dimension is aligned with the squeeze, while cracks lying parallel to the maximum stress feel little across their opening and stay open. Closed cracks stiffen the rock; open cracks keep it soft. A wave traveling along the maximum-stress direction crosses mostly closed, stiffened rock and runs fast; a wave traveling perpendicular to it crosses the surviving open cracks and runs slow. So the fast direction of the rock lines up with the maximum horizontal stress, and the slow direction lies across it.

An Illustrative Azimuthal Curve, and Where the Real Machinery Lives

Sketch the velocity against azimuth and it traces a simple two-lobed shape, fast one way, slow ninety degrees around, and fast again. A convenient illustrative form is V(\theta) = V_0\,[\,1 + A\cos 2(\theta - \theta_0)\,], with $\theta_0$ the maximum-stress azimuth and $A$ the strength of the anisotropy. Take $V_0 = 2.90$ km/s and $A = 0.04$: velocity peaks at 3.02 km/s along the maximum stress and dips to 2.78 across it, a spread of about 8 percent from fast to slow. The shape is illustrative and the widget says so; its point is the pattern, a velocity that is a function of compass direction, not just of depth. The full description, the Thomsen parameters that quantify weak anisotropy and the azimuthal amplitude and velocity signatures that let you measure the fast direction from surface seismic, is developed in the Seismic Modeling course: Thomsen anisotropy in its Part 7 and azimuthal and fracture signatures in its Part 8. Here the message is that stress writes a direction into the rock, and a repeat survey can watch that direction change. That closes the physics of pressure and time-lapse. The last section of this part asks the blunt engineering question underneath all of it: will the 4D signal actually beat the noise?