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

Deducing anisotropic drained constants from undrained for very heterogeneous porous media

At this point I have determined a data processing scheme that would provide all the drained constants for a poroelasticity system from measurements of the undrained constants. In the example of the preceding subsection, I needed to broaden the meaning of the undrained set of constants to include the Skempton $A_i$ coefficients, which were not needed in earlier parts of the paper. But they could nevertheless be computed from the information found earlier, since I did show how to compute all the $\beta_i$'s directly, and these coefficients provide just the information that I would need for determining these values from (57).

In realistic data collection situations, especially those involving field data, my previous assumptions concerning the nature and orientations of the constituent grains of the granular porous medium may sometimes - perhaps most times - be too idealized. Nevertheless, it is the case that the equations of poroelasticity never become any more complex than those shown here. What does change is the interpretation of the directional grain moduli. In the worst case scenario, equation (60) needs to be replaced by an equation of the same form, namely:

$$\frac{1}{3K_i^*} = s_{i1}^u + s_{i2}^u + s_{i3}^u - \beta_i(1-B).$$ (59)

Measurements are exactly as before, but the interpretation of the resulting constant estimator $K_i^*$ becomes that of an effective medium bulk modulus, i.e., one that is (or at least could be) dependent on the directions $i = 1,2,3$ of measurement. Effective medium theories for random polycrystals generally assume [see Berryman (2005)] that the anisotropic grains are perfectly randomly oriented. Of course, this may not be true in practice. But to do a better job of predicting the outcome of experiments in situations where grain orientations are not perfectly random, I need information about these deviations from perfect randomness. In the present context, the information would come in the form of these measured constants $K_i^*$. Some effort could then be expended in showing how such moduli might arise if the constituents' nature and volume fractions are known. But in the absence of such knowledge, these constants are sufficient to analyze other results of most experiments of interest in poroelasicity.

To provide different ideas about how important the anisotropy, and the random orientation of the constituents might be in a few cases, TABLES 1 and 2 show some quantitative examples based on results of Berryman (2005) and Ranganathan and Ostoja-Starzewski (2008a,b). TABLES 3 and 4, respectively, provide input data for the types of orthorhombic solids (Musgrave, 2003), and the results for the Voigt, Reuss, and directional measures of bulk moduli for these particular materials. Note the significant finding that the directional moduli do not have to stay within the values set by the Voigt and Reuss estimators.

Tables 5-9 show similar results for selected data taken from the compendium assembled by Simmons and Wang (1971).

1.2

TABLE 7.for the principal stiffness coefficients $c_{ij}$ for $i,j = 1,2,3$ and $c_{44}$ of cubic symmetry minerals: aluminum (Al), copper (Cu), magnesia (MgO), and spinel (MgAl$_2$O$_4$). All data from Simmons and Wang (1971) [entry numbers: 10089, 10385, 10902, and 11877, respectively], but re-expressed in units of GPa.

Stiffness	Aluminum (Al)	Copper (Cu)	Magnesia (MgO)	Spinel (MgAl$_2$O$_4$)
$c_{11}$	107.30	170.98	297.08	298.57
$c_{12}$	60.80	123.99	95.36	153.72
$c_{44}$	28.30	75.45	156.13	157.58

1.2

TABLE 8.for various measures of bulk modulus $K$ (Voigt, Reuss, and three partial-sum moduli) for cubic symmetry minerals: aluminum (Al), copper (Cu), magnesia (MgO), and spinel (MgAl$_2$O$_4$). All data from Simmons and Wang (1971) [see TABLE 7], while the expressions in the main text were used for all the computations. All moduli in units of GPa. Clearly, all the pertinent bulk moduli for each material are the same (i.e., $K_V = K_R = K_1 = K_2 = K_3$, even though these cubic symmetry minerals are not isotropic.

Bulk Modulus	Al	Cu	MgO	MgAl$_2$O$_4$
$K_V = K_R = \ldots$	76.3	139.65	162.6	202.00

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