Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media

The $\beta_i$ coefficients

I will now provide several results for the $\beta_i$ coefficients, and then follow the results with a general proof of their correctness.

In many important and useful cases, the coefficients $\beta_i$ are determined by

$$\beta_i = s_{i1}^d + s_{i2}^d + s_{i3}^d - \frac{1}{3K_R^g}.$$ (28)

Again, $K_R^g$ is the Reuss average of the grain modulus, since the local grain modulus is not necessarily assumed uniform here as mentioned previously. Equation (28) holds true for homogeneous grains, such that $K_R^g = K^g$. It also holds true for the case when $K_R^g$ is determined instead by (6). However, when the grains themselves are anisotropic, I need to allow again for this possibility, and this can be accomplished by defining three directional grain bulk moduli determined by:

$$\frac{1}{3\overline{K}_i^g} \equiv s_{i1}^g + s_{i2}^g + s_{i3}^g = s_{1i}^g + s_{2i}^g + s_{3i}^g,$$ (29)

for $i = 1,2,3$. The second equality follows because the compliance matrix is always symmetric. I call these quantities in (29) the partial grain-compliance sums, and the $\overline{K}_i^g$ are the directional grain bulk moduli. Then, the formula for (28) is replaced by

$$\beta_i = s_{i1}^d + s_{i2}^d + s_{i3}^d - \frac{1}{3\overline{K}_i^g}.$$ (30)

Note that the factors of three have been correctly accounted for because

$$\sum_{i=1,2,3} \frac{1}{3\overline{K}_i^g} = \frac{1}{K_R^g},$$ (31)

in agreement with (8). If the three contributions represented by (29) for $i = 1,2,3$ happen to be equal, then each equals one-third of the sum (31).

The preceding results are for perfectly aligned grains. If the grains are instead perfectly randomly oriented, then it is clear that the formulas in (28) hold as before, but now $K_R^g$ is determined instead by (8).

All of these statements about the $\beta_i$ are easily proven by considering the case when $\sigma_{11} = \sigma_{22} = \sigma_{33} = -p_c = -p_f$. In this situation, from (22), I have:

$$-e_{ii} = \left(s_{i1}^d + s_{i2}^d + s_{i3}^d\right)p_c + \beta_... ...(s_{i1}^g + s_{i2}^g + s_{i3}^g\right)p_f \equiv \frac{p_f}{3\overline{K}_i^g},$$ (32)

in the most general of the three cases discussed, and holding true for each value of $i = 1,2,3$. This is a statement about the strain $e_{ii}$ that would be observed in this situation, as it must be the same if these anisotropic (or inhomogeneous) grains were immersed in the fluid, while measurements were taken of the strains observed in each of the three directions $i = 1,2,3$, during variations of the fluid pressure $p_f$. Consider this proof to be a thought experiment for determining the coefficients, in the same spirit as those proposed originally by Biot and Willis [see Stoll (1974); Biot and Willis (1957)] for the isotropic and homogeneous case.

Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media