Azimuthal AVO (AVAz)

The Gradient That Depends on Compass Direction

Here the whole part pays off. A fractured HTI reservoir gives a reflection whose AVO gradient depends on the azimuth of the shot. Ruger wrote it compactly:

The intercept $R\_0$, the isotropic gradient $B\_{iso}$ and the curvature $C\_{iso}$ behave as in ordinary AVO. The new term is the anisotropic gradient $B\_{ani}$, which scales with crack density and fluid through the Hudson model and rides on $\\cos^2$ of the azimuth measured from the symmetry axis $\\phi\_{sym}$, the crack normal, ninety degrees from the fracture strike. A line shot ALONG the fractures stays in the isotropy plane and keeps the plain gradient $B\_{iso}$; a line shot ACROSS them feels the full $B\_{ani}$. The two share an intercept but fan apart as the offset grows, and the amplitude at a fixed angle traces an ellipse around the compass whose two axes lie on the strike and on the crack normal.

Ruger's expression for the anisotropic gradient is

with $k$ the average $V\_S/V\_P$ across the interface, $\\delta^{(V)}$ the Thomsen-style $\\delta$ of the fractured rock and $\\gamma^{(S)} \\approx \\Delta\_T/2$ its shear-wave splitting parameter. The two terms pull against each other. The normal weakness $\\Delta\_N$ makes $\\delta^{(V)}$ negative; the tangential weakness $\\Delta\_T$ makes $\\gamma^{(S)}$ positive. Gas-filled cracks keep the full normal weakness and the two largely cancel, so in this model $B\_{ani}$ stays below about 0.02 for gas. Brine props the cracks against closing, removes most of the normal weakness and leaves the shear term in charge: $B\_{ani}$ is about +0.04 at a crack density of 0.08. Either way it is small, a few hundredths at most against an isotropic gradient between about $-0.14$ and $-0.17$, which is why the azimuthal signal is a fine effect. At the angles a survey uses, the curvature term is not small either. Its $\\varepsilon^{(V)}$ comes from the normal weakness again and is negative, so the azimuthal swing $R\_{across} - R\_{along}$ is $\\big(B\_{ani} + \\tfrac12\\varepsilon^{(V)}\\tan^2\\theta\\big)\\sin^2\\theta$. At 30 degrees and a crack density of 0.05 the curvature wins for gas and the reflection is highest ALONG the strike, $-0.0035$; for oil and brine the gradient wins and it is highest across, $+0.0030$ and $+0.0059$. An exact plane-wave solve agrees on the side in every case on the controls, but gives gas nearly twice that swing, $-0.0063$, and oil less, $+0.0021$: the linearised gradient is low for gas-filled cracks.

Inverting Data Back to Geology

The star move is the inversion. Sample that azimuthal amplitude at the handful of azimuths a real survey provides, add realistic gauge noise, and fit a $\\cos(2\\phi)$ ellipse through the points. The axes of the fitted ellipse hand back the fracture strike, the very number the model was built with. Geology went in as a strike; rock physics and the wave turned it into a synthetic azimuthal amplitude; and a least-squares fit brings it back out as a strike. Turn the noise up and the recovery degrades gracefully, which is exactly why fracture surveys chase wide azimuths and strong contrasts: you need enough azimuthal signal to beat the gauge noise.

Two caveats close the argument. The fluid sets the size and the side of the azimuthal signal, and not in step with the velocity anisotropy two sections back: brine, which cut the P-wave velocity anisotropy, gives the largest anisotropic gradient here; gas-filled fractures put the highest amplitude on the strike instead of the crack normal; and oil-filled ones, between the two, give the smallest swing in the exact solution at every density on the controls, so they are the hardest to orient from P-wave amplitudes alone. The linearised model, low for gas, agrees on that only below a crack density of about 0.065. And the fitted axis alone carries a ninety-degree ambiguity between strike and its normal, resolved with the sign of the azimuthal term $B\_{ani} + \\tfrac12\\varepsilon^{(V)}\\tan^2\\theta$ (negative puts the highest amplitude on the strike, positive on the crack normal) or with shear data. That shear companion, geometry-only and fluid-blind, is the subject of the closing section.