Although pore geometry plays an important role in carbonates rock physics modeling, few studies have been done on its analytic relationship with other pore space properties like pore space stiffness. We propose an analytical workflow based on the differential effective medium (DEM) to estimate the elastic properties of carbonate rocks. Then, the validity of our results is cross-checked with the Xu and Payne model on a real carbonate dataset. This workflow establishes a direct and quantitative link between the pore geometry of carbonate rock with its other pore space properties such as Biot's coefficient and pore space stiffness. This relationship can be, furthermore, utilized in defining rock physics templates (RPTs) to investigate the role of pore geometry on the rock elastic properties. Furthermore, we extended the Biot-Gassmann-Krief (BGK) model through our proposed workflow by establishing a theoretical framework to relate the main components of the BGK model to the pore geometry usually estimated in the laboratory or empirically. This can help to investigate the impact of fluid substitution on each of these main components. Our investigation suggests that the higher the Biot and Gassmann coefficients, the rock is more sensitive to fluid substitution. Moreover, this analytical workflow has been employed to examine the role of selecting different rotational spheroids (i.e., oblate and prolate) on the modeled velocities. Our results show that the modeled velocities depend on this selection in a way that prolate pores are less sensitive to the variations in their pore aspect ratio compared with the oblate pores.