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

Heterogeneous grains

When the grains in a granular packing are no longer composed of elastically homogeneous and isotropic materials, or if they are homogeneous but anisotropic while nevertheless being distributed in a randomly oriented way in space, then -- as has been pointed out previously in Brown and Korringa (1975), Rice and Cleary (1976), and the work of others [e.g., Wang (2000)] -- I need to introduce a more general notation to deal with these circumstances.

Recall that the Reuss (1929) average of the grain bulk moduli for a heterogeneous medium with a distribution of grain types is given by:

$$\frac{1}{K_R^g} \equiv \sum_{m=1,\dots,n}\frac{v_m}{K_m},$$ (6)

where $v_m$ is the volume fraction (out of all the solid material present, so that $\sum_m v_m = 1$) of the $m$-th isotropic grain having bulk modulus $K_m$. This average should be distinguished from that of the Voigt (1928) average

$$K_V^g \equiv \sum_{m=1,\dots,n} v_m K_m,$$ (7)

which is known (Hill, 1952) to satisfy $K_V^g \ge K_R^g$. Furthermore, these two measures are also known (Hill, 1952; Voigt, 1928; Reuss, 1929) to satisfy $K_V^g \ge K_g^* \ge K_R^g$, where $K_g^*$ is the effective bulk modulus of an isotropic elastic composite consisting only of the minerals $m=1,\ldots,n$ in the same volume proportions given by the $v_m$ values. However, this fact actually is not pertinent here, as the only averages of this type that play a direct role in the poroelastic equations are always those of the Reuss-type, as will be shown in the further developments.

To clarify later usage of the same notation $K_R^g$, I emphasize here that when (or if) the grains in our assemblage are all anisotropic -- but nevertheless of the same type and oriented randomly in space -- then the pertinent average is again the Reuss average. But in this case the average is determined by the equation

$$\frac{1}{K_R^g} = \sum_{i,j=1,2,3} s_{ij}^g,$$ (8)

where the $s_{ij}^g$ for $i,j = 1,2,3$ are the principal components of the compliance matrix for the anisotropic grain material itself. It is easy to see that this must be the case by referring back to the equations above, specifically those requiring the suspension result $K_{susp}$. The formula as quoted in (2) was only written for the case of homogeneous grains. But if we generalize this formula slightly as:

$$K_{susp} \equiv \left[\frac{1-\phi}{K_R^g} + \frac{\phi}{K_f}\right]^{-1},$$ (9)

then we see that it holds equally true: (a) for homogeneous isotropic grains (when $K_R^g \equiv K^g$), (b) for heterogeneous volumes of isotropic grains [when $K_R^g$ is given by (6)], or (c) for anisotropic grains when they are randomly oriented in the fluid [and then $K_R^g$ is given by (8)]. In all these cases, I do assume that this mixture of grains and fluid is close to being a true suspension, by which we mean that individual grains are acted on similarly by changes in fluid pressure.

If the clumpings are loose enough, then the fluid can act equally on all the individual grains, and the result in (9) holds true regardless of the heterogeneity. However, if this is not the case, then there must be elastically distinct clumpings of grains forming solid composites locally - so the individual grains are no longer uniformly surrounded by the pore fluid. Then, each grain's fluid environment is different, due to welded contacts with other contiguous grains. I am assuming for the present purposes that such effects are negligible in the types of comparatively homogeneous porous media (on the meso- and macroscales, but not necessarily on the microscale) being studied here. In fact, some types of more heterogeneous systems can be treated, and some of these have already been studied (Berryman and Milton, 1991; Berryman and Pride, 2002) when the porous system is composed of just two distinct types of grain clumpings; however, I will not be discussing such double-porosity and/or multi-porosity effects in the present paper.

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