Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.