Backus Averaging

Part 9, Part 9: Layers, Shales, and Frequency

Learning objectives

  • Turn a stack of thin isotropic layers into one effective VTI medium with five stiffnesses via the Backus average
  • Read the two anchoring identities: identical layers return an isotropic medium, and C33 is exactly the series (Reuss) P-modulus of the stack
  • Put numbers on a 50/50 stiff-sand and shale stack: Thomsen epsilon 0.033, gamma 0.104, delta -0.042
  • See that the shear contrast dominates layer anisotropy, so gamma is the largest parameter and delta can go negative

From Five Layers to Five Stiffnesses

Part 9.1 left us with a stack of thin isotropic beds that a seismic wave sees as one medium. Backus, in 1962, wrote down exactly what that one medium is. Take layers whose thicknesses are much less than a wavelength, each described by its Lame parameters lambda\lambda and mu\mu and its thickness fraction. The trick is that stresses and strains combine differently along the layers than across them: quantities continuous across the bed boundaries are averaged one way, quantities that add up through the stack are averaged another. The result is a single medium with vertical symmetry, a VTI medium, described by five stiffnesses: C11,C33,C13,C44C_{11}, C_{33}, C_{13}, C_{44}44, and C66C_{66}66. C33C_{33}33 and C44C_{44}44 are harmonic (series) averages of the layer moduli, because loading across the beds compresses them one behind another like springs in series; C66C_{66}66 is a plain arithmetic average, because shearing along the beds loads them side by side; and C11C_{11}11 and C13C_{13}13 mix the two, each carrying a harmonic average inside arithmetic ones. That asymmetry is the whole origin of the anisotropy.

Two identities pin the method down and are worth carrying as sanity checks. First, if every layer is identical the five stiffnesses collapse back to a single isotropic medium and there is no anisotropy at all, exactly as physics demands. Second, C33C_{33}33, the stiffness governing a P-wave travelling vertically across the layers, is exactly the series (Reuss) average P-wave modulus of the stack. The wave crossing the beds is only as stiff as the softest springs in the chain allow, and Backus reproduces that limit precisely.

Backus averagingthin isotropic layerssandshaleBackusone VTI mediumVp 3.535along layers, fast3.424across, slowε 0.033 γ 0.104 δ -0.042Layering alone makes the rock anisotropic; the shear contrast makes gamma the largest.

A Sand-Shale Stack in Numbers

Make it concrete with a 50/50 stack of a stiff quartz sand (bulk modulus 20 GPa, shear 12, density 2.30) and a shale (bulk 15.01, shear 4.96, density 2.31, the Ogbon-1 shale read through its logged Vp of 3.06 km/s with a Castagna shear estimate). Backus returns C33=27.02C_{33} = 27.0233=27.02 GPa and C11=28.81C_{11} = 28.8111=28.81 GPa, so the medium is stiffer along the layers than across them: a P-wave runs at 3.535 km/s horizontally and only 3.424 km/s vertically, a spread of about 3.3 percent from layering alone. Converting the five stiffnesses to Thomsen parameters, the compact anisotropy report a seismologist actually uses, gives epsilon=0.033\epsilon = 0.033, gamma=0.104\gamma = 0.104, and delta=−0.042\delta = -0.042. The parameter epsilon\epsilon measures the P-wave speed-up along the layers, gamma\gamma the shear-wave anisotropy, and delta\delta the near-vertical behaviour that governs moveout and small-angle AVO.

Shear Contrast Runs the Show

Look at which parameter is largest. Here gamma=0.104\gamma = 0.104 dwarfs epsilon=0.033\epsilon = 0.033, and delta\delta has come out negative. That ordering is not an accident of this particular pair; it is the signature of layering. The anisotropy of a layered stack is driven mostly by the contrast in shear modulus between the layers, and the sand-shale contrast in mu\mu (12 against about 5) is far larger than the contrast in P-modulus, so the shear-controlled gamma\gamma leads. The sign of delta\delta is a subtler business of competing terms, and it commonly runs negative for such stacks, which is why a layered shale can produce non-hyperbolic moveout that a purely isotropic model would misread. Shrink the contrast and the anisotropy nearly vanishes: a nearly velocity-matched brine soft sand against the same shale gives epsilon=−0.004\epsilon = -0.004 and gamma=0.015\gamma = 0.015, almost isotropic, with epsilon\epsilon even edging negative because the Poisson contrast pulls against the shear contrast. The robust statement is that gamma\gamma is never negative for layering; the others can be. What these Thomsen numbers do to a seismic reflection, how they bend an AVO curve and shift a moveout, belongs to the Seismic Modeling course, Part 7, which owns the anisotropic reflectivity. Here we have shown where the numbers come from. The next section asks the companion question: shales are anisotropic even when you cannot see any layering, so how much of shale anisotropy is Backus layering and how much is the rock's own fabric.

References

  • Backus, G. E. (1962). Long-wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research, 67(11), 4427-4440.
  • Thomsen, L. (1986). Weak elastic anisotropy. Geophysics, 51(10), 1954-1966.

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