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

Deducing drained constants from undrained: Heterogeneous grains and pores

I was able to deduce $K_R^d$ from our knowledge of $K_R^u$, $K_R^g$, $K_f$, and $\phi$ in subsection 2.2. But even though I am still assuming the system is isotropic, I have now introduced some additional degrees of freedom by permitting the grains and pores to be heterogeneous. It is clear that I cannot deduce $K_R^d$ if I just have the same amount of information as before. In particular, it does seem fairly straightforward to measure $K_R^g$, as I have already described its meaning in the earlier discussion and even given formulas for it -- if I have information about the constituents and their volume fractions, or alternatively about the principal components of elastic compliance and/or stiffness matrices. But I have another variable now, which is the pore modulus $K_R^\phi$, and this bulk modulus is not so easy either to model or to measure directly (Lockner and Stanchits, 2002). However, by adding one more piece of information -- namely the second Skempton coefficient $B = p_f/p_c$, which is a fact that should typically be known in poroelastic systems -- then it turns out that I can solve for both $K_R^d$ and $K_R^\phi$. Again, I assume that $K_R^g$ and $K_R^u$ are known. But now I also assume that $B$ is also known experimentally. Working through the algebra, I find that

$$K_R^d = \frac{1-B}{1/K_R^u - B/K_R^g}$$ (20)

[which is a rearrangement of $K_R^u = K_R^d/(1-\alpha_R B)$], and similarly that

$$\frac{1}{K_R^\phi} = \frac{1}{K_f} - \left(\frac{1-B}{\phi B}\rig... ...f} - \left(\frac{1}{\phi B}\right)\left(\frac{1}{K_R^u}-\frac{1}{K_R^g}\right).$$ (21)

In (21), I used the previous result (20) for $K_R^d$ to simplify the final formula.

These forms are very useful for many applications in poroelasticity, but so far they apply only to the fully isotropic case. I show next that a very similar set of formulas applies to the anisotropic cases under consideration. I am able to attain greater clarity at this point by switching to the more general anisotropic problem, where it can seen more easily how poroelastic reciprocity comes directly into play.

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