The natural-convection flow and associated heat transfer in a fluid-saturated porous medium have been investigated using the generalized porous medium approach for a dome-shaped enclosure. Many new features have been predicted with the connective heat transfer and the shape of the top dome cover. The solutions are obtained for a wide range of Darcy and Rayleigh numbers for different offsets and eccentricities of the top dome covers. The detailed parametric study reveals that there is a significant change in heat transfer rate when the offset is between 0.2 and 0.4. Different shapes of conic section, such as circular, elliptical, parabolic, and hyperbolic are used for the top dome cover, and their effects on natural convection and heat transfer rates are studied.