Summary
The classic differential effective medium (DEM) theory can be used to determine the elastic properties of the porous medium, but the final elastic properties of multiple-porosity rock depend on the added order of the different pore-type inclusions. This paper presents a differential effective medium model for multiple-porosity rock and derives the analytical solutions of the bulk and shear moduli for dry rock. The validity of these analytical approximations is tested by integrating the full DEM equation numerically. The analytical formulae give good estimates of the numerical results over the whole porosity range. The analytical formulae have been used to predict the elastic moduli for the sandstone experimental data by assuming that the porous rock contains dual-porosity of both cracks and pores. The results show that they can accurately predict the variations of elastic moduli with porosity for dry sandstones.
Introduction
The elastic properties of the rock depend on the microscopic structure of pore system significantly. Most of rocks usually have two or even more than two different pore types, such as pore, crack, cavity, etc., whose complex pore system makes the relationship between the velocity and porosity of the rock highly scattered (Sayers, 2008; Baechle, et al., 2008). Therefore, it needs to establish a multiple-porosity rock physical model for characterizing the elastic modulus of porous rock varying with porosity accurately.
Effective medium theory is often used to study the elastic properties of porous rock, such as Kuster-Toksöz theory and differential effective medium theory. Kuster and Toksöz (1974) derived expressions for bulk and shear moduli of multiple-porosity rock by using wave scattering theory, in which the effects of elasticity, volume content and pore shape of inclusions are taken into account. But Kuster-Toksöz theory does not consider the interaction between different pores, and requires that the ratio of porosity to aspect ratio has to be much smaller than 1. Kuster-Toksöz theory has been shown to violate the upper and lower Hashin-Shtrikman bounds in some cases (Berryman, 1980). The differential effective medium (DEM) theory models two-phase composites by incrementally adding inclusions to the matrix phase, and is applied to the determination of the effective elastic properties of porous rocks dry and saturated by fluid through the numerical solution of differential equations (Berryman, 1980; Norris, 1985; Zimmerman, 1985; Berryman, et al., 2002). The DEM has the property that it can never violate rigorous bounds comparing with the Kuster and Toksöz theory (Berryman, 1980; Berryman and Berge, 1996; Berryman et al., 2002). However, since the ordinary differential equations for bulk and shear moduli are coupled, it is more difficult to integrate them to yield accurate analytical formulae for the bulk and shear moduli. Therefore, the behavior of the bulk and shear moduli can only be accurately simulated through the numerical solution of differential equations. Moreover, the effective elasticities of porous medium depend on the order in which the incremental additions of the inclusions, i.e. the pores or cracks with different aspect ratios, are done.
