Andrew John Callegari

A generalized differential effective medium theory

Abstract A generalization of the Differential Effective Medium approximation (DEM) is discussed. The new scheme is applied to the estimation of the effective permittivity of a two phase dielectric composite. Ordinary DEM corresponds to a realizable microgeometry in which the composite is built up incrementally through a process of homogenization, with one phase always in dilute suspension and the other phase associated with the percolating backbone. The generalization of DEM assumes a third phase which acts as a backbone. The other two phases are progressively added to the backbone such that each addition is in an effectively homogeneous medium. A canonical ordinary differential equation is derived which describes the change in material properties as a function of the volume concentration φ of the added phases in the composite. As φ→ 1, the Effective Medium Approximation (EMA) is obtained. For φ

Effective‐medium theories for two‐phase dielectric media

Two different effective‐medium theories for two‐phase dielectric composites are considered. They are the effective medium approximation (EMA) and the differential effective medium approximation (DEM). Both theories correspond to realizable microgeometries in which the composite is built up incrementally through a process of homogenization. The grains are assumed to be similar ellipsoids randomly oriented, for which the microgeometry of EMA is symmetric. The microgeometry of DEM is always unsymmetric in that one phase acts as a backbone. It is shown that both EMA and DEM give effective dielectric constants that satisfy the Hashin–Shtrikman bounds. A new realization of the Hashin–Shtrikman bounds is presented in terms of DEM. The general solution to the DEM equation is obtained and the percolation properties of both theories are considered. EMA always has a percolation threshold, unless the grains are needle shaped. In contrast, DEM with the conductor as backbone always percolates. However, the threshold in EMA can be avoided by allowing the grain shape to vary with volume fraction. The grains must become needlelike as the conducting phase vanishes in order to maintain a finite conductivity. Specifically, the grain‐shape history for which EMA reproduces DEM is found. The grain shapes are oblate for low‐volume fractions of insulator. As the volume fraction increases, the shape does not vary much, until at some critical volume fraction there is a discontinuous transition in grain shape from oblate to prolate. In general, it is not always possible to map DEM onto an equivalent EMA, and even when it is, the mapping is not preserved under the interchange of the two phases. This is because DEM is inherently unsymmetric between the two phases.

Differential effective medium theory of sedimentary rocks

We show that both the electrical and acoustic properties of fluid‐saturated sedimentary rocks can be described within the unified framework of differential effective medium theory. Calculations based on the differential effective medium picture of rock microstructure yield predictions of sonic travel times and acoustic attenuation in good agreement with experimental data. In particular, the theory shows that the large frequency peak in attenuation and its associated velocity dispersion observed in sandstones are characteristic of a composite system containing fluid‐filled microcracks.

Consistent theoretical description for electrical and acoustic properties of sedimentary rocks

We present a differential effective medium theory which is capable of providing a consistent description for both the electrical and acoustic properties of sedimentary rocks. Besides the correct prediction of dc and finite‐frequency electrical behaviors for sandstones, calculations based on the differential effective medium picture of rock microstructure are shown to yield sonic travel times and acoustic attenuation in good agreement with experimental data. Physical interpretation of the various rock characteristics within our theoretical framework leads to the interesting picture of a rock as a fluid‐solid composite system that is in the vicinity of a percolation threshold.