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Differential effective medium (DEM) 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 effective medium 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 (Cleary et al., 1980; Norris, 1985; Avellaneda, 1987) -- and one version of the self-consistent approach (Korringa et al., 1979; Berryman, 1980a,b; Milton, 1985) is the other. This fact, together with the associated analytical capabilities (including ease of computation and flexibility of application), provides strong motivation to study the predictions of both of these schemes and the differential scheme in particular. 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 the inclusions are sufficiently sparse that they do not form a single connected network throughout the composite, it is most appropriate to use the Differential Effective Medium (DEM) to model their elastic behavior (Berge et al., 1993). Assume that the host material has moduli Km and $\mu_m$,while the inclusion material has moduli Ki and $\mu_i$. Then, the effective bulk and shear moduli (indicated as such by the asterisks) of the composite are parametrized by K*(y) and $\mu^*(y)$ where the volume fraction of the inclusion phase is y. The equations governing the changes in these constants are then well-known to be

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

(1-y)d^*(y)dy = [_i-^*(y)]Q^*i,   where the scalar factors, P*i and Q*i, will be explained in the following paragraph, y is porosity which equals inclusion volume fraction here, and the subscript i again stands for inclusion phase. We assume that the reader is somewhat familiar with this approach, and will therefore not dwell on its derivation, which is easily found in many places including, for example, Berryman (1992). These equations are typically integrated starting from porosity y = 0 with values K*(0) = Km and $\mu^*(0) = \mu_m$,which are assumed here for modeling purposes to be the mineral moduli values for the single homogeneous solid constituent. Integration then proceeds from y=0 to the desired highest value $y=\phi$ (the porosity of the sample), or possibly over the whole range to y=1 for some purposes of analysis. When integrating this way, we might imagine the result is, for example, 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 $\mu^*(\phi_0) = \mu_s$. 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).

The factors P*i and Q*i appearing in (DEMK) and (DEMmu) are the so-called polarization factors for bulk and shear modulus (Eshelby, 1957; Wu, 1966). 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 usually have 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) and Mavko et al. (1998).

Because it is relevant both to low porosity granites and to sandstones having equant (i.e., close to spherical) porosity as well as flat cracks, the case we consider here is that of penny-shaped cracks, where

P^*i = K^* + 43_iK_i+43_i+^*   and

Q^*i = 15[1 + 8^*4_i+ (^*+2^*) + 2K_i + 23(_i+^*) K_i+43_i+^*],   with $\alpha$ ($0 < \alpha < 1$)being the crack (oblate spheroidal) aspect ratio,

^* ^*[(3K^*+^*)/(3K^* +4^*)],   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 $\mu_i/\mu_m << 1$ and allows Ki/Km << 1, with these approximations being made before any 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 quite inappropriate for the bulk modulus when the inclusion phase is a gas such as air (for then the ratio Ki/Km << 1) or for the shear modulus when the inclusion phase is any fluid (for then $\mu_i \equiv 0$), as the formulas become singular in these limits. This is why the penny-shaped crack model is commonly used instead for cracked rocks.

In general the DEM equations (DEMK) and (DEMmu) 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. Later in the paper, we will show results obtained from integrating the DEM equations numerically.

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