Cracks to Stiffness

Part 8, Part 8: Fractures and Rock Physics

Learning objectives

See a crack as a compliant flaw with a direction
Relate crack density to normal and tangential weaknesses
Watch the across-crack velocity fall with crack density
Recognise a crack set as the physical cause of HTI

A Flaw That Knows Which Way You Push

Part 7 described anisotropy as a fact about velocity. Part 8 builds it from a cause: cracks. A crack is a compliant flaw, and its compliance has a direction. Push across the crack faces and they close easily, so the rock is soft in that direction. Push along the faces and the crack barely notices, so the rock stays stiff. A single crack is already directional; a whole set of parallel cracks makes the rock directionally soft at the seismic scale, which is exactly the HTI symmetry we met before, now with a physical origin.

The linear-slip model summarises a crack set by two dimensionless numbers. The normal weakness $\Delta_N$ N measures how much softer the rock is across the cracks; the tangential weakness $\Delta_T$ T does the same for shear sliding along them. Both grow with the crack density $e$ , essentially the number and size of cracks per unit volume, and both vanish when $e=0$ .

The Velocity Gap Is the Anisotropy

The consequence is direct. The P velocity travelling across the cracks falls as $V_{p0}\sqrt{1-\Delta_N}$ , while the velocity along the cracks is almost untouched. Raise the crack density and the two pull apart. That growing gap between the along-crack and across-crack velocities is the anisotropy, no longer an abstract ellipse but the summed compliance of many small flaws.

This is the bridge the whole part is built on. A geologist counts fractures; a rock physicist turns that count into weaknesses; a modeller turns the weaknesses into an anisotropic velocity field that a wave equation can propagate. The next section makes the middle step explicit with the Hudson model, mapping crack density straight into the Thomsen parameters of an equivalent HTI medium.