Analytical approximations of bulk and shear moduli for dry rock based on the differential effective medium theory

摘要

Differential effective medium theory has been applied to determine the elastic properties of porous media. The ordinary differential equations for bulk and shear moduli are coupled and it is more difficult to obtain accurate analytical formulae about the moduli of dry porous rock. In this paper, in order to decouple these equations we first substitute an analytical approximation for the dry-rock modulus ratio into the differential equation and derive analytical solutions of the bulk and shear moduli for dry rock with three specific pore shapes: spherical pores, needle-shaped pores and penny-shaped cracks. Then, the validity of the analytical approximations is tested by integrating the full differential effective medium equation numerically. The analytical formulae give good estimates of the numerical results over the whole porosity range for the cases of the three given pore shapes. These analytical formulae can be further simplified under the assumption of small porosity. The simplified formulae for spherical pores are the same as Mackenzie's equations. The analytical formulae are relatively easy to analyse the relationship between the elastic moduli and porosity or pore shapes and can be used to invert some rock parameters such as porosity or pore aspect ratio. The predictions of the analytical formulae for experimental data show that the formulae for penny-shaped cracks are suitable to estimate the elastic properties of micro-crack rock such as granite, they can be used to estimate the crack aspect ratio while the crack porosity is known and also to estimate the crack porosity evolution with pressure if the crack aspect ratio is given.