Anisotropy and azimuthal attributes: fractures and stress

Types of anisotropy

• VTI (Vertical Transverse Isotropy): rotationally symmetric about a VERTICAL axis. Dominant in shales (fine-scale horizontal layering at grain scale) and in fine-bedded sandstones. Vp is faster along bedding than across. No azimuthal variation (the rose is a CIRCLE), but Vp depends on ZENITH ANGLE from vertical.
• HTI (Horizontal Transverse Isotropy): rotationally symmetric about a HORIZONTAL axis, typically normal to aligned vertical fractures. Vp depends on AZIMUTH: fastest along the fracture strike (no fractures crossed), slowest perpendicular. The velocity rose is an OVAL, indistinguishable from an ellipse at weak anisotropy, with the long axis along fracture strike.
• Orthorhombic: VTI + HTI combined. Both horizontal layering AND aligned vertical fractures. Most realistic model for fractured sedimentary reservoirs. Has TWO axes of symmetry (bedding normal + fracture strike).
• Lower-symmetry (monoclinic, triclinic): rarely modeled; assumes even more complex fracture networks or folding-induced fabrics. Industry standard is orthorhombic unless there’s clear evidence for more.

Exercise, build the anisotropy from the cracks up

The widget below is not a drawing of anisotropy, it is a calculation of it. A crack density of aligned vertical cracks in an isotropic background is turned into an effective medium using Hudson's compliances and Schoenberg's linear-slip stiffnesses, and every velocity you see comes from putting those stiffnesses through the exact Christoffel equation. Five views look at the same rock in five ways.

• Start in The crack set. A crack density e = 0.05 of 1 m cracks implies a spacing of about 6.4 m, and that is the spacing drawn. Each arrow is a ray, and the label counts the cracks that ray actually crosses in the picture: zero along strike, nine straight across. Read the velocities beside the counts, then note something the old cartoon of this section got wrong: EVERY direction is slower than the unfractured 3500 m/s. Cracks add compliance, and compliance cannot speed a rock up.
• Switch to Velocity rose. Radius is proportional to velocity on a scale that starts at zero, so the flattening you see is the anisotropy at true proportion. Drag strike and the oval rotates with it. Drag density e to zero and the oval closes to the blue reference circle exactly.
• Now change Fill from dry to brine, still in the rose view. The oval relaxes almost back to a circle while the cracks are all still there. Brine resists being squeezed shut, so it kills the normal compliance and with it nearly all of the P-wave azimuthal anisotropy.
• Go to Shear-wave splitting and change the fill again. The delay does not move by a microsecond. A fluid carries no shear stress, so it cannot stop a crack from sliding: the tangential compliance is blind to the fill. This is why splitting is the robust fracture-density measurement, and why the two observations together tell you something neither can alone.
• Open VTI: ε, δ and γ, which is a different rock: a layered shale rather than a cracked sand. Drag ε and only the horizontal end of the P curve moves. Drag δ and the near-vertical curvature moves, taking the moveout velocity Vnmo with it while the horizontal end stays put. That is the whole reason the two parameters exist separately.
• In the same view, set δ equal to ε. The solid P curve lands exactly on the dashed elliptical reference and the SV bump vanishes. Pull them apart again and the anellipticity comes back; that gap is what stops far offsets flattening when a processor assumes ellipticity.
• Finish in AVAz inversion. The dashed curve is a real least-squares fit to the noisy samples, and the strike it reports is computed from that fit. Now drag incidence down from 30° toward 5° and watch the two recovered numbers part company: the axis stays about right, while the fitted amplitude, which is what fracture density is read from, roughly doubles. Short offsets tell you which way the fractures run, not how many there are.

Thomsen parameters

Thomsen (1986) parameterized weak anisotropy with three dimensionless numbers for VTI media:

• ε (epsilon): relative P-wave anisotropy between bedding-parallel and bedding-normal. Typical shales: ε ≈ 0.1-0.25. High-fracture-density HTI: ε ≈ 0.05-0.15.
• δ (delta): the NEAR-vertical P-wave anisotropy parameter (governs the curvature at small zenith angles). Controls depth errors from isotropic imaging. Typical δ ≈ 0-0.15.
• γ (gamma): S-wave (shear) anisotropy. Parameter in converted-wave and multicomponent data.

In HTI media the same parameters are DEFINED relative to the symmetry axis (fracture normal), often noted as ε_V, δ_V, γ_V. Orthorhombic uses SIX Thomsen-style parameters to capture both VTI and HTI contributions. These numbers are what inversion actually solves for.

AVAz: amplitude variation with azimuth

The practical workhorse of HTI inversion. At a given reservoir interface, the reflection amplitude at a FIXED OFFSET varies with propagation azimuth if the rock below is HTI. The amplitude is a SINUSOID with period 180° (fractures look the same forward or back):

A(φ) = A₀ + B · cos(2(φ − φ_fast))

where A₀ is the azimuthally averaged amplitude, B is the anisotropy magnitude, and φ_fast is the fast-axis azimuth (= fracture strike for simple HTI). Inversion per voxel: fit the 2φ sinusoid to amplitude vs azimuth. Outputs: B (fracture density proxy) and φ_fast (fracture strike).

Requires WIDE-AZIMUTH data: traditional narrow-azimuth marine streamer surveys don’t sample enough of the azimuth space for reliable AVAz. Ocean-bottom-node, wide-azimuth towed-streamer, and land 3D (multi-component vibrator arrays) are all AVAz-capable.

Practical applications

• Fractured reservoirs: the strongest use case. Carbonate reservoirs, tight sandstones, and fractured basement all have aligned-fracture networks that HTI inversion maps directly. Fracture density (from ε) predicts permeability; fracture strike (from φ_fast) guides horizontal-well azimuth selection.
• Stress-direction mapping: aligned fractures respond to the current stress field. Open fractures align with MAXIMUM HORIZONTAL STRESS (σH); compressed fractures close normal to σH. AVAz fast-axis = current σH azimuth, directly useful for wellbore-stability and frac-design.
• Imaging correction: below a VTI overburden (shale caps, clay packages), isotropic migration mis-positions reflectors in depth. Adding ε and δ to the imaging model removes these errors, TOC (top-of-carbonate) and other critical events image much more accurately.
• Shear-wave splitting (converted waves): in HTI media, a shear wave splits into FAST (polarized along fracture strike) and SLOW (normal to strike) components. Time delay between them is another direct fracture-density measurement.
• Unconventional plays (shale + frac): identify “sweet spots” where natural fractures already exist. Drive completion design: hit fracture corridors with multi-stage fracs oriented transverse to the natural-fracture strike.

When anisotropy matters and when it doesn’t

• Must-model situations: (1) Imaging below a VTI overburden (most sedimentary basins, shales are almost always VTI). (2) Fractured reservoirs where fractures are PRODUCTION-CONTROLLING. (3) AVAz inversion for stress/fracture maps. (4) Anisotropic velocity analysis for accurate depth conversion.
• Optional (but improving): (1) Conventional reservoirs with weak anisotropy (ε < 0.05) and no fractures of interest. (2) Basins with well-calibrated legacy isotropic velocity models.
• Required cost: anisotropic processing adds ∼15-30% to the compute time for pre-stack depth migration. Wide-azimuth acquisition is ∼30-50% more expensive than narrow-azimuth. But the payoff is cleaner imaging + new information (fracture/stress maps) that isotropic can’t deliver.
• Common mistake: using narrow-azimuth data for AVAz. Doesn’t work. The math requires samples from MANY azimuths at the same CDP; narrow-azimuth just doesn’t have them. AVAz projects always start with “do we have the data?” before “can we do the analysis?”.