Thermally driven flows in a two-dimensional rectangular cavity filled with a fluid-saturated porous medium are considered when the applied temperature difference is perpendicular to the gravity vector. The flow depends on two non-dimensional parameters, the Darcy-Rayleigh number A and the cavity aspect ratio h (height/length). Steady motion is generated by maintaining the vertical sidewalls at different constant temperatures and the present study is concerned with the limit of large aspect ratio, h $\rightarrow \infty $, where nonlinear convective flow occurs near each end of the cavity. This flow is analysed by asymptotic methods for small and large values of A and is computed numerically for other values of A. Predictions of the heat transfer across the cavity are obtained and are compared with the results of full numerical simulations.