Using a recently introduced numerical technique known as a lattice-Boltzmann method, we numerically investigate immiscible two-phase flow in a three-dimensional microscopic model of a porous medium and attempt to establish the form of the macroscopic flow law. We observe that the conventional linear description of the flow is applicable for high levels of forcing when the relative effects of capillary forces are small. However, at low levels of forcing capillary effects become important and the flow law becomes nonlinear. By constructing a two-dimensional phase diagram in the parameter space of nonwetting saturation and dimensionless forcing, we delineate the various regions of linearity and nonlinearity and attempt to explain the underlying physical mechanisms that create these regions. In particular, we show that the appearance of percolated, throughgoing flow paths depends not only on the relative concentrations of the two fluids but also on a dimensionless number that represents the ratio of the applied force to the force necessary for nonwetting fluid to fully penetrate through the porous medium. Finally, we fit a linear model to our simulation data in the high-forcing regions of the system and observe that an Onsager reciprocity holds for the viscous coupling of the two fluids. ¿ American Geophysical Union 1993