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Differential effective medium theory (Bruggeman, 1935; Cleary et al., 1980; Walsh, 1980; Norris, 1985; Avellaneda, 1987) takes the point of view that a composite material may be constructed by making infinitesimal changes in an already existing composite. There are only two schemes known at present that are realizable, i.e., that have a definite microgeometry associated with the modeling scheme. The differential scheme is one of these (Norris, 1985; Avellaneda, 1987). This fact provides a strong motivation to study the predictions of this theory because we can have confidence that the results will always satisfy physical and mathematical constraints, such as the Hashin-Shtrikman bounds (Hashin and Shtrikman, 1961; 1962).

When inclusions are sufficiently sparse that they do not form a single connected network throughout the composite, it is appropriate to use the Differential Effective Medium (DEM) to model their elastic behavior (Berge et al., 1993). If the effective bulk and shear constants of the composite are K*(y) and G*(y) when the volume fraction of the inclusion phase is y, then the equations governing the changes in these constants are

(1-y)dK^*(y)dy = [K_i-K^*(y)]P^*i   and

(1-y)dG^*(y)dy = [G_i-G^*(y)]Q^*i,   where y is also porosity in the present case and the subscript i stands for inclusion phase. These equations are typically integrated starting from porosity y = 0 with values K*(0) = Km and G*(0) = Gm, which are assumed here to be the mineral values for the single homogeneous solid constituent. Integration then proceeds from y=0 to the desired highest value $y=\phi$, or possibly over the whole range to y=1. When integrating this way, we imagine the result is simulating cracks being introduced slowly into a granite-like solid. The same procedure can be used for a sandstone-like material assuming this medium has starting porosity $y = \phi_0$ with $K^*(\phi_0) = K_s$ and $G^*(\phi_0) = G_s$. Integration then proceeds from $y = \phi_0$ to y = 1. This introduction of crack (or soft) porosity into a material containing spherical (or stiff) porosity is conceptually equivalent to the porosity distribution model of Mavko and Jizba (1991). For simplicity, we will treat the granite-like case here, but the changes needed for other applications are not difficult to implement, and are treated specifically in Appendix A.

The factors P*i and Q*i appearing in (DEMK) and (DEMG) are the so-called polarization factors for bulk and shear modulus. These depend in general on the bulk and shear moduli of both the inclusion, the host medium (assumed to be the existing composite medium * in DEM), and on the shapes of the inclusions. The polarization factors have usually been computed from Eshelby's well-known results (Eshelby, 1957) for ellipsoids, and Wu's work (Wu, 1966) on identifying the isotropically averaged tensor based on Eshelby's formulas. These results can be found in many places including Berryman (1980b, 1995) and Mavko et al. (1998).

The special case of most interest to us here is that for penny-shaped cracks, where

P^*i = K^* + 43G_iK_i+43G_i+^*   and

Q^*i = 15[1 + 8G^*4G_i+ (G^*+2^*) + 2K_i + 23(G_i+G^*) K_i+43G_i+^*],   with $\alpha$ being the crack (oblate spheroidal) aspect ratio, $\gamma^* = G^*[(3K^*+G^*)/(3K^* +4G^*)]$, and where the superscript * identifies constants of the matrix material when the inclusion volume fraction is y. This formula is a special limit of Eshelby's results not included in Wu's paper, but apparently first obtained by Walsh (1969). Walsh's derivation assumes that Ki/Km << 1 and Gi/Gm << 1, and makes these approximations before making assumptions about smallness of the aspect ratio $\alpha$. By taking these approximations in the opposite order, i.e., letting aspect ratio be small first and then making assumptions about smallness of the inclusion constants, we would obtain instead the commonly used approximation for disks. But this latter approximation is actually inappropriate for the bulk modulus when the inclusion phase is air or gas (for then the ratio Ki/Km << 1) or for the shear modulus when the inclusion phase is any fluid (for then $G_i \equiv 0$), as the formulas become singular in these limits. This is why the penny-shaped crack model is commonly used instead for rocks.

In general the DEM equations (DEMK) and (DEMG) are coupled, as both equations depend on both the bulk and shear modulus of the composite. This coupling is not a serious problem for numerical integration, and we will show results obtained from integrating the DEM equations numerically later in the paper. But, the coupling is a problem in some cases if we want analytical results to aid our intuition. We will now present several analytical results for both bulk and shear modulus, and then compare these results to the fully integrated DEM results later on.

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