This paper presents a series of numerical simulations dealing with the problem of natural convection flows and associated heat transfer in an enclosure filled with a fluid-saturated porous medium. The analysis is based on the finite element technique and incorporates the Brinkman-extended Darcy model for an oval enclosure., The numerical results obtained for a modified Rayleigh number, Ra, Darcy number, Do, offset, E, and eccentricity, e, are presented and discussed. The numerical predictions,for a square enclosure compared well with published data. It is found that any increase in Da or Ra results in a higher fluid velocity that is responsible for shifting the core of the flow. Moreover, at higher ovality (E = 0.5), asymmetric flow is observed even at the lower range of Rayleigh number (Ra less than or equal to 20), which may be attributed to the effect of curved isothermal wall.