We test a number of empirical models, such as the Wyllie time average and Raymer?s velocity-porosity equation, as well as physic based effective-medium models, such as the self-consistent and differential effective medium theory, against three published datasets. Two of these datasets are velocity measurements in artificial composites made of metal particles embedded in epoxy resin in a range of particle concentration. The third dataset is velocity in water-saturated carbonate. We find that for certain ranges of inclusion concentration and porosity, the empirical equations (Wyllie?s and Raymer?s) fail. At the same time, an effective-medium model, such as DEM, appears to be consistently valid if the aspect ratio is selected appropriately and then held constant for the entire concentration or porosity range. This aspect ratio can be found by calibrating the theory to the data at a single data point and then holding it constant in the entire concentration/porosity range. The intention of this work is to set a consistent rigorous foundation for modeling of the elastic properties of sediment with inclusions, such as carbonate, with the ultimate goal of consistent interpretation of log and seismic data for rock properties and texture.
A composite with isotropic matrix and randomly embedded and oriented inclusions is isotropic. Theories for the effective elastic moduli of such a composite are of two general types: (1) upper and lower bounds and (2) exact solutions. The narrowest bounds are due to Hashin and Shtrikman (1963). Mal and Knopoff (1967) obtained effective properties for a two-phase medium with the assumptions that the matrix is solid, the inclusions are spherical, much smaller than the wavelengths of the propagating waves, and that interactions between inclusions are negligible. Kuster and Toksöz (1974a) derived theoretical expressions for long wavelengths based on scattering theory. Both spherical and oblate spheroidal inclusions were considered in their calculations. When comparing experimental data for the case of a liquid matrix with solid inclusions, Kuster and Toksöz (1974b) concluded that the observed data could be fit either by their model or by the Reuss (1929) lower bound appropriate for the elastic moduli of a suspension. More recent are the differential effective medium theory (DEM) and self consistent methods (SC). DEM (Bruggeman, 1935; Walsh, 1980; Norris, 1985; Avellaneda, 1987; Berryman, 1992) assumes that a composite material may be constructed by making infinitesimal changes in an already existing composite. It is relevant to modeling the elastic properties of a porous medium with inclusions in a wide porosity range. SC uses mathematical solutions for the deformation of isolated inclusions, but the interaction of inclusions is approximated by replacing the background medium with as-yet-unknown effective medium (Mavko, et al., 1998). Berryman (1992) proposed three single scattering approximations for estimating the effective elastic properties of composite materials: (a) the average Tmatrix approximation; (b) the coherent potential approximation; and (c) DEM. Devaney and Levine (1980) proposed another model based on a self consistent formulation of the multiscattering theory. Their approach assumed that the inclusions are spherical and that the wavelengths are longer than the size of the inclusions.