Appendix B: Hill's Equation and Heterogeneous Porous Media

One very common approximation made in studies of partially and patchy saturated porous media (Norris, 1993; Mavko et al., 1998; Johnson, 2001) is based on an assumption that the estimates are being made over a small enough region that it is reasonable to take the shear modulus of the porous frame as constant, even though the bulk modulus over the same small region may vary. Then, when Gassmann's results apply locally, the shear modulus satisfies $\mu_d = \mu_u$, and so remains constant throughout this same region regardless of the distribution of fluids in the pores. When these assumptions are valid, Hill's equation (3) may be used to compute the effective bulk modulus K^*, regardless of anisotropy or of how many constituents might be present. Furthermore, Hill's equation will apply equally to the drained K_d^* and undrained K_u^* bulk moduli of such a poroelastic system; K_n for the layers must be substituted accordingly for the drained and undrained cases.

This approximation based on Hill's equation is very appealing for applications because of its analytical beauty and overall simplicity, but its use in heterogeneous media has never been given a rigorous justification. In particular, the assumption of variable bulk modulus in a heterogeneous system having constant shear modulus is surely one worthy of careful consideration. It seems more likely (at least to me) that the variations in the bulk modulus in an earth system will be mimicked by the shear modulus and, therefore, that the proposed method is in truth an oversimplification of this complex problem.

The model system presented here (i.e., the random polycrystal of porous laminates) offers one means of checking whether this use of Hill's equation might be justified or not.

It turns out that, when N = 2, Hill's equation (3) can be inverted to give $\mu$ as a functional of K^* (Milton, 1997). The result is given by  

$$\mu = \frac{3K_1K_2}{4K_r}\left(\frac{K^* - K_r}{K_v - K^*}\right),$$ (30)

where  

$$K_v = \sum_{n=1}^2 f_nK_n\qquad\hbox{and}\qquad K_r = \left[\sum_{n=1}^2 \frac{f_n}{K_n}\right]^{-1}.$$ (31)

So I can do two calculations based on the results presented here for heterogeneous systems. We can compute effective shear moduli $\mu_d^{\rm eff}$ and $\mu_u^{\rm eff}$by taking the self-consistent values to be the true values of the drained and undrained K^*, and layer values of K_d⁽ⁿ⁾ and K_u⁽ⁿ⁾ as the values for K₁ and K₂. The volume fractions are those already used in these calculations. So everything is known and the computations are straightforward. We want to check whether the resulting values of effective shear moduli $\mu_d^{\rm eff}$ and $\mu_u^{\rm eff}$ computed this way are approximately constant and/or approximately equal to each other. If they are, then Hill's equation, although not rigorously appropriate in these systems, nevertheless could be capturing some of the observed behavior. If this is not true, then the results would be showing that great care should be exercised in using these formulas for analyzing real data.

My results are illustrated in Figure 12. I find that $\mu_d^{\rm eff} \simeq \mu_u^{\rm eff}$. However, except for the volume fractions near 50%, the values of both $\mu^{\rm eff}$'s are very different from the actual shear moduli of the random polycrystals of porous laminates model. The $\mu^{\rm eff}$'s are high when the $\mu^*$'s are low, and vice versa. This observation is a very strong negative result, showing that large errors in analysis can be introduced for systems such as these that are very heterogeneous in shear.

On the positive side, it is also clear from Figure 12 that if the spread of layer $\mu$'s is nonzero but small, then the use of Hill's equation can be well justified. The error in shear estimates will never be greater than the spread in the layer shear modulus values, so if this is a small (though nonzero) number, then the errors will be finite but correspondingly small.

 

Fig12
Figure 12 Illustrating computations of an effective shear modulus obtained by inverting Hill's equation for drained ($\mu_d^{\rm eff}$)and for undrained patchy saturation ($\mu_u^{\rm eff}$) conditions. Model parameters are the same as in Figure 9 for patchy saturation. For comparison the curves for self-consistent shear moduli $\mu_d^*$ and $\mu_u^*$ from Figure 10 are replotted here.