A Differential Effective Medium Model of Multiple‐Porosity Rock and Its Analytical Approximations for Dry Rock

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 order of adding the different pore-type inclusions due to the lack of DEM equations for multiple-porosity rock. This paper first derives the differential equations of both Zimmermann's and Norris's versions for multiple-porosity rock from the Kuster-Toksoz theory. The elastic moduli predicted by the DEM equations of Norris's version never violate Hashin-Shtrikman bounds while those predicted by the DEM equations of Zimmermann's version violate bounds in some cases. Then, we derive analytical solutions of the bulk and shear moduli for dry rock from the differential equations of Norris's version by applying an analytical approximation for dry-rock modulus ratio, in order to decouple these equations. 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. We also apply nonlinear global optimization algorithm to find the best estimate for the pore volume percentage of both cracks and pores by minimizing the error between theoretical predictions and experimental measurements based on the dual-porosity DEM analytical model. The inversion results of the sandstone experimental data show that there is no direct correlation between the crack volume percentage and clay volume.