We present a fast technique for modeling convective displacements that are dominated by large-scale reservoir heterogeneities. The novel feature of our method is the integration of any 1D solution with periodically updated streamtubes or streamlines to simulate displacements in two and three dimensions. We use streamtubes in two dimensions and streamlines in three dimensions. We construct approximate solutions using two to five orders of magnitude less computation time than by conventional simulation. Our approach allows us to decouple the physics describing the displacement from the size of the grid used to model the reservoir geology. In addition, because 1D solutions are often analytical, or can be obtained numerically using higher-order methods, the resulting 2D and 3D solutions are free of numerical diffusion. We present examples in two and three dimensions for a variety of displacement mechanisms and compare our results with conventional finite-difference solutions showing excellent agreement. Because of its speed, our method becomes particularly effective for estimating the uncertainty in forecasting reservoir performance resulting from the uncertainty in the description of the reservoir geology.