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Theoretical Formulae

We first compare our results to three of the most commonly used theoretical methods: Hashin-Shtrikman bounds [Hashin and Shtrikman 1963], the self-consistent approximation (SCA) [Hill 1965,Budiansky 1965,Berryman 1980], and the differential effective medium (DEM) approach [Berryman 1992,Mavko 1998].

A specification of the volume fraction and constituent moduli allows the calculation of rigorous upper and lower bounds for the elastic moduli of any composite material. The so-called Hashin-Shtrikman bounds [Hashin and Shtrikman 1963] are given by

Upper and lower bounds are computed by interchanging the moduli of the solid and fluid components. In the case where one phase has zero elastic moduli, the lower bound becomes zero, and so only the upper bound is meaningful.

A commonly-used effective medium theory, the differential (DEM) method is constructed by incrementally adding inclusions of one phase into the second phase with known constituent properties. DEM does not treat each constituent symmetrically, but defines a preferred host material. From the composite host medium, () at some porosity value $\phi$ is known. One then treats () as the composite host medium and ( + d ) as the effective constant after a small proportion d / (1 - ) of the composite host has been replaced by inclusions of the second phase. For a solid matrix host, the coupled system of ordinary differential equations for the moduli is given by [Berryman 1992]

with initial conditions K*(0) = K s and µ*(0) = µs and where P* and Q*are shape dependent geometric factors for inclusions of the second phase in a background medium with effective moduli K* and µ*, as given in several texts (see e.g., Table 4.9.1 of [Mavko 1998]). For the current work we use the geometric factors for spherical grains (pores).

In the self consistent model (SCA) of [Hill 1965] and [Budiansky 1965] the host medium is assumed to be the composite itself. The equations of elasticity are solved for a spherical inclusion embedded in a medium of unknown effective moduli. The effective moduli are then found by treating Kscm, µscm as tunable quantities. The result is given in a general form [Berryman 1980] by

In the present work we use the geometric factors for spherical pores and a number of granular inclusion shapes. The variation with granular shape had a minimal effect (~1%) on the predictions, so we report results for spherical pores and grains only. The indices to P and Q note the inclusions of fluid "*fi" and solid "*si into a background medium of effective moduli K* and µ*. As for the DEM equations the solution for the effective bulk moduli is found iteratively.

The SCA produces a single formula in which all components are treated equally, with no material distinguished as the host to others. Such a symmetric formula has been thought to be more appropriate in complex aggregates like granular rocks and has been shown [Berge 1993] to accurately predict the mechanical behavior of a sintered glass bead sample.

We compare the three theories to our numerical predictions and experimental data for both dry and water saturated rock in Figure 7. We note that none of the theoretical methods results in a satisfactory fit to the experimental data. In contrast, the numerical results are in excellent agreement with the experimental data. The SCA theory gives a much better fit to the experimental data than either the DEM or the Hashin-Shtrickman upper bound. This is consistent with the observation of Berge et al. [ Berge 1993 ] that the SCA should more accurately predict the elastic properties of granular rocks. However, the numerical prediction is far superior to any of the theoretical estimates. This conclusion is in accord with recent results of [Roberts and Garboczi 2000] where it was shown that neither bounds, SCA nor DEM successfully predict the properties of sintered granular materials.

Figure 7: Comparison of the simulation results to a range of theories used to predict the moduli of porous rocks. The curves include predictions for the (a) dry and (b) water-saturated bulk modulus and (c) the shear modulus. All theories overestimate the data for all porosities. The SCA gives the best theoretical fit to the data as expected from (Berge 1993 Berge et al., 1993), but is much poorer than the numerical prediction from the tomographic data.

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