Hudson and Linear Slip

The Bridge: A Fracture Count In, A Model Out

The previous section gave the intuition; the Hudson (and linear-slip) model gives the equation. Feed it a crack density and it returns the three anisotropy parameters of an equivalent HTI medium, the medium whose smooth, seismic-scale properties match a rock full of small aligned cracks. This is the single most useful step in fracture seismology, because it lets a geologist's fracture count become a modeller's velocity field.

The three parameters, $\epsilon^{(V)}$, $\delta^{(V)}$, and $\gamma^{(V)}$, are all negative for a gas-filled fracture set: the crack-normal direction is the slow one, so the anisotropy pulls the parameters below zero. They deepen smoothly as crack density rises and vanish when there are no cracks. A key detail: $\gamma^{(V)}$ comes only from the tangential (shear) weakness, so it responds to the fracture geometry alone.

What the Modeller Receives

The right panel turns those parameters into what a modelling code actually consumes: a velocity ellipse in the vertical plane, fast along the crack faces and slow across them. Raise the crack density and it flattens. Nothing here is qualitative any more. Given a crack density, you have a numeric anisotropic velocity model you can hand to the wave equation, and the synthetic seismogram of a fractured reservoir follows directly.

That is the fit-for-purpose payoff of the rock-physics detour. You did not need to mesh thousands of individual cracks into a finite-difference grid, which would be impossible at seismic scale. You replaced them with three numbers that carry their averaged effect exactly where the wave feels it. The next section adds the missing ingredient, the fluid inside the cracks, and shows how $\gamma^{(V)}$'s independence from it becomes a fluid detector.