The macroscopic properties and structure of the flow of two immiscible fluids through Fontainebleau sandstone are studied by numerical simulation. The pore space geometry was obtained by X-ray microtomography (Kinney et al. 1993) and the numerical simulations were performed by a new lattice-gas cellular automaton method (Olson & Rothman 1995). We first validate the numerical method by showing that the drag on a cubic array of spherical drops matches theoretical predictions. As a further test, we present a comparison between computed relative permeability and experimental measurements on the same rock. We then present a study of fluid–fluid coupling; we find that it is significant, and that it appears to be reciprocal: the flux of one fluid due to forcing on the other is the same, regardless of which fluid is forced. Lastly, we characterize the complexity and organization of the flow by means of a statistical parameter, the skewness of the distribution of local velocities.