Backus averaging

Backus (1962) presented an elegant method of producing the effective constants for a thinly layered medium composed of either isotropic or anisotropic elastic layers. This method applies either to spatially periodic layering or to random layering, by which we mean either that the material constants change in a nonperiodic (unpredictable) manner from layer to layer or that the layer thicknesses might also be random. For simplicity, we will assume that the physical properties of the individual layers are constant and isotropic. [For applications to porous earth materials, we implicitly make the typical assumptions of spatial stationarity within these layers as well as scale separation -- i.e., the sizes of the pores are much smaller than the thickness of the individual layers in which they reside.] The key idea presented by Backus is that these equations can be rearranged into a form where rapidly varying (in depth) coefficients multiply slowly varying stresses or strains.

The derivation has been given many places including Schoenberg and Muir (1989) and Berryman (1999a). Another illuminating derivation has been given recently by Milton (2002). We will not repeat any of the derivations here. The final results will be expressed in terms of averages $\left<Q\right\gt$, where the brackets $\left<\cdot\right\gt$ surrounding a variable Q(z) indicate the volume average (or, equivalently, the linear average with depth in the vertically layered medium under consideration) of the quantity Q. It follows that the anisotropy coefficients in equation (1) are then related to the layer parameters by the following well-known expressions:

$$\begin{eqnarray} c = \left<{{1}\over{\lambda+2\mu}}\right\gt^{-1}, \end{eqnarray}$$ (8)

$$\begin{eqnarray} f = c\left<{{\lambda}\over{\lambda+2\mu}}\right\gt, \end{eqnarray}$$ (9)

$$\begin{eqnarray} l = \left<{{1}\over{\mu}}\right\gt^{-1}, \end{eqnarray}$$ (10)

$$\begin{eqnarray} m = \left<\mu\right\gt, \end{eqnarray}$$ (11)

$$\begin{eqnarray} a = {{f^2}\over{c}} + 4m - 4\left<{{\mu^2}\over{\lambda+2\mu}}\right\gt, \end{eqnarray}$$ (12)

and

 

<I>bI> = <I>aI> - 2<I>mI>. (13)

When the layering is fully periodic, these results may be attributed to Bruggeman (1937) and Postma (1955), while for more general layered media including random media they should be attributed to Backus (1962). The constraints on the Lamé parameters $\lambda$ and $\mu$ for each individual layer are $0 \le \mu\le \infty$ and $-{2\over3}\mu\le \lambda\le \infty$. Although, for physically stable materials, shear modulus $\mu$ and bulk modulus $K = \lambda+ {2\over3}\mu$ must both be nonnegative, these relations mean that $\lambda$ (and also Poisson's ratio $\nu$) may be negative (but nevertheless bounded below, since $\nu \ge -1$, and $\lambda \ge -2\mu/3$). Large fluctuations in $\lambda$for different layers are therefore entirely possible, in principle, but may or may not be an issue for any given region of the earth.

Large fluctuations in $\mu$ are also possible, and the Backus averaging technique is fully capable of handling all such fluctuations properly. But, if these fluctuations are too large, then the weak anisotropy assumption of Thomsen's original work (Thomsen, 1986) will be violated and some care must be taken when writing approximate equations. We do not at any point assume weak anisotropy in this paper [except equations (5) and (28)], since the shear behavior we are trying to study will be shown to depend on the presence of strong anisotropy in this sense. We will also find it useful to develop alternatives to some of Thomsen's formulas in order to deal with the strong anisotropy that arises in our analysis.

One very important fact known about the Backus averaging equations (Backus, 1962) is that they reduce to isotropic results with a=c, b=f, and l=m, if the shear modulus is a constant ($= \mu$) -- regardless of the behavior of $\lambda$.This fact is also very important for applications involving partial and/or patchy saturation (Mavko et al., 1998; Johnson, 2001). Furthermore, this fact is closely related to the well-known bulk modulus formula of Hill (1963) for isotropic composites having uniform shear modulus, and also to the Hashin-Shtrikman bounds (Hashin and Shtrikman, 1961), which can be used to provide an elementary proof of Hill's equation. Nevertheless, this limit will not be of much interest to us here except as a boundary condition on the results obtained. Furthermore, one of the main purposes of the paper is to show how deviations from these limiting and rather restictive results affect the predictions of the referenced work on partial and patchy saturation.