A theoretical study of buoyancy-driven flow and natural convection heat transfer in a fluid saturated porous medium in conditions of thermal stratification has been numerically studied in this thesis. A large number of investigations dealing with such types of flows have been reported in the literature due to a variety of applications in science, engineering and industry. As far as the theoretical investigations of flows in porous media are concerned, various flow models have been used by researchers such as the classical Darcy, Brinkman and Forchheimer models. Unsteady natural convection flow and heat transfer inside a right angle trapezoidal enclosure filled with a fluid saturated porous medium has been investigated numerically. The left and bottom walls of the trapezoidal enclosure are kept at constant temperature Th and To respectively with Th > To, while the temperature for the right wall. varies as To = To + By where B is the gradient of Tc and the upper wall is insulated. To solve the governing partial differential equations we convert them into a non-dimensional form using a suitable transformation of variables. The simulation is carried out through the pde solver COMSOL Multiphysics which uses the Galerkin weighted residual based finite element method. Particular efforts have been focused on the effects of time, Rayleigh number, thermal stratification parameter, porosity of the porous medium, different solid matrix of the porous medium (glass ball, aluminium foam and sandstone), various fluids (water, kerosene and engine oil) on the average Nusselt numbers, streamlines and isotherms. The obtained results show that heat transfer rate is an increasing function of the Rayleigh number and decreasing function of the other considered key parameters which are thermal stratification parameter, porosity and aspect ratio. It is also found that the average Nusselt number for aluminum foam with engine oil is much higher than that of the others types of flows in porous media.