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 isotropic. The key idea presented by Backus is that these equations can be rearranged into a form where rapidly varying coefficients multiply slowly varying stresses or strains.

The derivation has been given many places including Schoenberg and Muir (1989) and Berryman (1999a). One illuminating derivation given recently by Milton (2002) will be followed here, with the main difference being that we assume the layering direction is z or 3. We break the equation down into $3\times 3$ pieces so that

$$\begin{eqnarray} \left( \begin{array} {c} \sigma_{11} \ \sigma_{22} \ \sigma... ...in{array} {c} e_{33} \ e_{23} \ e_{31} \ \end{array}\right) \end{eqnarray}$$ (8)

and

$$\begin{eqnarray} \left( \begin{array} {c} \sigma_{33} \ \sigma_{23} \ \sigma... ...n{array} {c} e_{33} \ e_{23} \ e_{31} \ \end{array}\right), \end{eqnarray}$$ (9)

where the $3\times 3$ matrices are

$$\begin{eqnarray} A_{11} = \left(\begin{array} {ccc} \lambda + 2\mu & \lambda & 0... ... 0 & 0 \ 0 & 2\mu & 0 \ 0 & 0 & 2\mu \ \end{array}\right). \end{eqnarray}$$ (10)

Noting that the variables $\sigma_{11}$, $\sigma_{22}$, $\sigma_{12}$,e₃₃, e₂₃, and e₃₁ are fast variables in the layers, and all the remaining variables are slow (actually constant), it is advantageous to rearrange these equations so the slow variables multiply the elastic parameter matrices and are all on one side of the equations, while the fast variables are all alone on the other side of the equations. Then, it is trivial to perform the layer averages, since they depend only on the (assumed known) values of the elastic parameters in the layers and are multiplied by constants. Having done this, we can then transform back into the standard forms of (8) and (9) with the stresses and strains now reinterpreted as the overall values, and find the following relationships (where the star indicates the effective property of the layered system):

$$\begin{eqnarray} A_{33}^* = \left<A_{33}^{-1}\right\gt^{-1}, \end{eqnarray}$$ (11)

$$\begin{eqnarray} A_{13}^* = \left(A_{31}^*\right)^T = \left<A_{13}A_{33}^{-1}\right\gt A_{33}^*, \end{eqnarray}$$ (12)

and

$$\begin{eqnarray} A_{11}^* = \left<A_{11}\right\gt + A_{13}^*\left(A_{33}^*\right)^{-1}A_{31}^* - \left<A_{13}A_{33}^{-1}A_{31}\right\gt. \end{eqnarray}$$ (13)

The brackets $\left< x \right\gt$ indicate the volume (or equivalaently the one-dimensional layer) average of the quantity x in the simple layered medium under consideration. It follows that the anisotropy coefficients in equation (1) are then related to the layer parameters by the following expressions:

$$\begin{eqnarray} c = \left<{{1}\over{\lambda+2\mu}}\right\gt^{-1}, \end{eqnarray}$$ (14)

$$\begin{eqnarray} f = c\left<{{\lambda}\over{\lambda+2\mu}}\right\gt, \end{eqnarray}$$ (15)

$$\begin{eqnarray} l = \left<{{1}\over{\mu}}\right\gt^{-1}, \end{eqnarray}$$ (16)

$$\begin{eqnarray} m = \left<\mu\right\gt, \end{eqnarray}$$ (17)

$$\begin{eqnarray} a = {{f^2}\over{c}} + 4m - 4\left<{{\mu^2}\over{\lambda+2\mu}}\right\gt, \end{eqnarray}$$ (18)

and

 

<I>bI> = <I>aI> - 2<I>mI>. (19)

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, $\mu$ and the bulk modulus $K = \lambda+ {2\over3}\mu$ must both be nonnegative, $\lambda$ (and also Poisson's ratio) may be negative. 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.

One very important fact that is known about these 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, it 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).