Anisotropy and azimuthal attributes: fractures and stress

Types of anisotropy

• VTI (vertical transverse isotropy): rotational symmetry about a vertical axis. Horizontal layering finer than the wavelength and aligned clay minerals make most shales VTI. Velocity depends on the angle from the vertical and not on azimuth: in map view every direction is alike.
• HTI (horizontal transverse isotropy): rotational symmetry about a horizontal axis, typically the normal to a set of aligned vertical fractures. Velocity depends on azimuth: fastest along the fracture strike, where a wave crosses no fracture, and slowest across them for dry or gas-filled fractures. The horizontal velocity rose is an oval close to an ellipse at weak anisotropy, its long axis along the strike.
• Orthorhombic: vertical fractures in a layered rock, with three mutually perpendicular symmetry planes. It is the usual model of a fractured sedimentary reservoir.
• Lower symmetries (monoclinic, triclinic): two fracture sets at an oblique angle, or tilted fabrics. They are rarely modelled; orthorhombic is the working standard unless the data demand more.

Thomsen’s parameters

Thomsen (1986) described a weakly anisotropic VTI medium by its vertical velocities $V_{P0}$ and $V_{S0}$ and three dimensionless numbers built from its stiffnesses $C_{ij}$:

• $\varepsilon$ fixes the horizontal P velocity, $V_P(90^\circ) = V_{P0}\sqrt{1 + 2\varepsilon}$. Shales commonly have $\varepsilon$ between about 0.05 and 0.3.
• $\delta$ controls the P velocity near the vertical, and with it the moveout velocity of a flat reflector, $V_{\mathrm{nmo}} = V_{P0}\sqrt{1 + 2\delta}$. It is usually smaller than $\varepsilon$ and can be negative.
• $\gamma$ does for SH what $\varepsilon$ does for P: $V_{SH}(90^\circ) = V_{S0}\sqrt{1 + 2\gamma}$.

To first order in the three parameters, the P velocity at angle $\theta$ from the vertical is $V_P(\theta) \approx V_{P0}(1 + \delta\sin^2\theta\cos^2\theta + \varepsilon\sin^4\theta)$, so $\delta$ shapes the near-vertical curve and $\varepsilon$ the horizontal end. When $\delta = \varepsilon$ the P wavefront is an ellipse, and a reflection’s moveout is an exact hyperbola. Their difference, as the anellipticity

is what bends the far offsets away from the hyperbola (Alkhalifah and Tsvankin 1995). In Figure 8.3 a shale with $\varepsilon = 0.20$ and $\delta = 0.10$ has $V_{\mathrm{nmo}}$ = 3834 m/s and $\eta$ = 0.083, and its reflection from 2000 m arrives 24.5 ms ahead of the hyperbola at 4000 m offset. A short-spread velocity analysis sees only $V_{\mathrm{nmo}}$: it cannot separate $V_{P0}$ from $\delta$, so converting time to depth with stacking velocities puts reflectors too deep by the factor $\sqrt{1 + 2\delta}$, about $\delta$ for small values. The far offsets measure $\eta$; the vertical velocity has to come from a well.

Orthorhombic media need more: in Tsvankin’s (1997) notation, seven dimensionless parameters ($\varepsilon^{(1)}, \varepsilon^{(2)}, \delta^{(1)}, \delta^{(2)}, \delta^{(3)}, \gamma^{(1)}, \gamma^{(2)}$) besides the two vertical velocities.

Cracks as an HTI medium

Aligned cracks add compliance to the rock, most of it across their faces. In the linear-slip picture (Schoenberg and Sayers 1995) a crack set adds two dimensionless weaknesses: a normal weakness $\Delta_N$, the cracks’ resistance to closing, and a tangential weakness $\Delta_T$, their resistance to sliding. For penny-shaped cracks of density $e = Na^3/V$ Hudson’s (1981) theory gives, to first order (Bakulin, Grechka and Tsvankin 2000), with $g = V_S^2/V_P^2$ of the background,

where $K$ grows with the fluid’s bulk modulus over the crack aspect ratio. A stiff fluid such as brine resists closing the crack and drives $\Delta_N$ toward zero; no fluid carries shear, so $\Delta_T$ is the same dry, gas-filled or brine-filled. Hudson’s theory is first order in $e$ and meant for dilute cracks, $e$ well below about 0.1; Figure 8.3 stops at 0.08, and its dry cracks of density 0.050 already take nearly a third off the stiffness across them. The equivalent Thomsen parameters, defined about the vertical in the plane of the crack normal, are

That split is the most useful fact in fracture seismology. In Figure 8.3, dry cracks of density 0.050 make the horizontal P velocity fall from 3436 m/s along the strike to 2922 m/s across it, a 15.0 % azimuthal anisotropy; gas-filled, 7.1 %; brine-filled, only 1.9 %, and the slowest direction moves to about 45° from the strike, because what is left is the tangential weakness acting through $\delta^{(V)}$. A vertical shear wave splits into a fast wave polarised along the strike and a slow one polarised across it, and that splitting, 6.2 % here, is the same for all three fills. P-wave azimuthal anisotropy is therefore a measure of crack density and fill together; shear-wave splitting measures crack density whatever the fill. Every direction stays slower than the uncracked rock: cracks only add compliance.

AVAz: amplitude variation with azimuth

Below an HTI layer the P-P reflection coefficient at a fixed incidence angle varies with azimuth. Rueger (1998) linearised it for an isotropic rock over an HTI rock:

with $\phi$ measured from the crack normal, $\gamma^{(S)} \approx -\gamma^{(V)} = \Delta_T/2$ the shear-wave splitting parameter (positive for cracks), and $R_0$, $B_{\mathrm{iso}}$ and $C_{\mathrm{iso}}$ the ordinary intercept, gradient and curvature of Part 5 for the rock as it is along the strike. At each incidence the azimuthal part is a sinusoid with period 180°, plus a small $\cos 4\phi$ part that a fit over evenly spread azimuths ignores, and an AVAz inversion fits $a + b\cos 2\phi + c\sin 2\phi$ to the amplitudes at many azimuths. What the fit returns is an axis pair 90° apart and an amplitude $2\sqrt{b^2 + c^2} = |B_{\mathrm{ani}} + \tfrac{1}{2}\varepsilon^{(V)}\tan^2\theta|\sin^2\theta$. Which member of the pair is the strike depends on the sign of that azimuthal term, and that has to come from a model or from other data. The gradient’s two terms pull against each other: the slow shear wave polarised across the cracks lowers the shear contrast in the plane of the crack normal and raises the gradient there, while the normal weakness inside $\delta^{(V)}$ lowers it. For the dry cracks of Figure 8.3 they nearly cancel and $B_{\mathrm{ani}}$ is small and negative; filled with gas or brine the normal weakness shrinks and $B_{\mathrm{ani}}$ turns positive. The curvature term matters as soon as the angle grows. Its $\varepsilon^{(V)}$, the parameter that flattens the velocity rose, is $-0.13$ for the dry cracks, so at 30° it outweighs the gradient about five times and the reflection is highest along the strike; for gas-filled cracks it cancels the gradient near 31°; brine all but removes it, and the reflection is highest across the cracks. The theory is linear, and for gas-filled cracks its gradient is 27 to 32 % below an exact plane-wave solve of the same interface, so the figure puts their change of sign at 30 to 31° where the exact solve puts it at 33 to 36°; for densities of 0.058 and up at 31 to 34° it picks the wrong axis, though the exact swing there is at most 0.0023, under twice what the noise alone fits.

Two cautions follow from the formula. First, the azimuthal AVO signal depends on the fill, and not in step with the velocity rose: in Figure 8.3, at 30° incidence, cracks of density 0.050 give 0.0067 dry, highest along the strike, 0.0070 brine-filled, highest across them, and 0.0002 gas-filled, less than the noise alone fits, while the P anisotropy falls from 15.0 % dry through 7.1 % gas to 1.9 % brine. Neither attribute alone measures crack density; shear-wave splitting, which ignores the fill, comes closer. Second, the signal scales with $\sin^2\theta$, so near offsets measure it badly against fixed noise: for the brine-filled cracks, with the figure’s noise, the recovered strike scatters by 4° rms at 30° incidence but 30° at 12°, where the worst draws pick the wrong member of the pair, and the fitted amplitude there reads about 30 % high on average. Azimuthal analysis needs wide-azimuth data with long offsets in every direction: ocean-bottom nodes or cables, multi- or wide-azimuth towed streamers, or a well-sampled land survey. Narrow-azimuth streamer data cannot support it.

Fractures or stress?

An azimuthal attribute reports a fast direction, and two different things put a fast direction into a rock. Open aligned fractures do it, and fractures stay open most easily when they strike parallel to the present maximum horizontal stress $\sigma_H$. Stress does it on its own as well: every rock holds compliant grain contacts and microcracks, and those normal to the maximum compressive stress close while those parallel to it stay open, so even an unfractured rock becomes faster along $\sigma_H$ (Sayers and Kachanov 1995). Both mechanisms commonly give the same fast azimuth, along $\sigma_H$, so azimuth alone cannot tell them apart. A cemented or mineralised fracture set oblique to today’s stress may be geologically important yet seismically almost invisible.

What separates them is everything except the azimuth. Stress-induced anisotropy weakens with depth and effective stress and changes when production changes the pore pressure; open fractures persist and their P-wave signature changes with the fluid that fills them, as the brine case of Figure 8.3 shows, while their shear-wave splitting does not. In practice the seismic maps are calibrated against wells: image logs for fracture strike and aperture, borehole breakouts and drilling-induced tensile fractures for the stress direction, and production for whether the fractures carry fluid.

Practical applications

• Fractured reservoirs: carbonates, tight sandstones and fractured basement, where fractures control flow. The fit’s axis maps the strike that guides well azimuth; density estimates come from calibrated amplitudes or, more robustly, from shear-wave splitting.
• Stress direction: the fast azimuth often follows $\sigma_H$, which sets the orientation of hydraulic fractures (they open against the least stress, $\sigma_h$, so they strike along $\sigma_H$) and the directions of wellbore breakout.
• Imaging: below a VTI overburden, isotropic migration misplaces reflectors and mis-ties wells in depth; $\varepsilon$ and $\delta$ in the migration velocity model, calibrated at wells, remove most of the error.
• Shear-wave splitting: from multicomponent surface data (converted waves) or VSPs, the polarisation of the fast shear wave gives the strike and the delay between the two gives a fluid-independent measure of crack density along the path.
• Unconventional plays: natural fracture corridors as sweet spots, and the stress direction that sets how stimulated fractures grow from a lateral.

When anisotropy matters and when it does not

• Must be modelled: depth imaging below shale-rich overburdens; fractured reservoirs where fractures control production; any azimuthal analysis for fractures or stress; depth conversion that must tie wells.
• Often negligible: weakly anisotropic conventional reservoirs ($|\varepsilon|, |\delta| \lesssim 0.05$) without fractures of interest, and areas where well-calibrated isotropic velocity models already tie.
• Cost: anisotropic depth migration and tomography add computation and parameters to estimate, and wide-azimuth, long-offset acquisition costs more than narrow-azimuth. What it buys is correct depths and information, fracture and stress maps, that isotropic processing cannot deliver.
• Common mistake: running AVAz on narrow-azimuth data, or reading a fitted amplitude as fracture density without checking the incidence range, the noise and the sign of the azimuthal term.