Abstract

There are two points of view with regard to the relationship between the dry-framework modulus ratio of bulk to shear modulus and porosity. One view, assumed by most empirical models such as the critical porosity model, is that the dry-framework modulus ratio is independent of porosity. However, this assumption is sometimes in disagreement with experiments. Another view supported by the differential effective-medium (DEM) theory is that the dry-framework modulus ratio depends on porosity. Ordinary differential equations (ODE) for bulk and shear moduli are coupled, and there is no analytical formula for the modulus ratio of dry porous rock. By assuming that the difference between the polarization factors of dry inclusions for bulk and shear modulus within the ODE has an almost linear relationship with the effective modulus ratio, an analytical solution for the dry-rock modulus ratio is derived from the ODE. The above linear relationship characterized by intercept a and gradient b is confirmed by the numerical results calculated from the analytical formulas for the three specific pore shapes: spherical pores, needle-shaped pores, and penny-shaped cracks. Then the validity of this assumption is tested by integrating the full DEM equation numerically. The analytical approximation gives a good estimate of the numerical results for the cases of the three given pore shapes. The dry-frame modulus ratio is monotonic with respect to porosity, and its monotonicity depends on the mineral-grain modulus ratio and the pore shape. The analytical formula can be simplified further to some popular models such as the critical porosity model by setting constants a and b to specific values. Predictions of the analytical formula compare favorably with real and synthetic sandstone data and granite data measured in the laboratory.

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