Percolation theory, it is often argued, requires a minimum nonwetting phase saturation of approximately 15% for hydrocarbon migration; but field measurements usually fail to find residual oil on the pathway taken during hydrocarbon migration from a source to a reservoir. We use numerical simulations and experiments to show that percolation theory, when applied to the process of oil invading a pore space, leads to the conclusion that field-scale saturations in secondary migration can be on the order of 1% or less. Our capillary invasion experiments on Berea Sandstone and numerical simulations of fluid invasion on networks show that the minimum saturation for flow of the nonwetting phase decreases as the reciprocal square root of the linear size. This result applies to cyli drical samples that have height equal to diameter. Our numerical simulations include the effects of sample aspect ratio, buoyancy, and pore-size distribution. All the numerical results reduce to a single relation between threshold saturations and sample size normalized by a function of the pore-size distribution. We find that a pore-size distribution that has many more small pores than large pores, as is typical in rocks and soils, substantially reduces the influence of buoyancy on threshold saturations. The common practice of using uniform (flat) pore- size distributions to simulate rocks overestimates the effects of buoyancy. Experiments show that for larger samples, the permeability at threshold for flow is higher than conventionally expected. In a typical geological situation, this h gh permeability is large enough to carry oil to a reservoir over geologic time at saturations of less than 1% of the pore volume. Our laboratory permeability and saturation experiments confirm the expectations of percolation theory, including the fractal dimension of the first invasion path. Size-dependent saturation is the prediction of invasion percolation theory. The percolation model is applicable to the secondary migration problem, as well as to migration of gas in reservoirs, migration of gas or fluids from radioactive waste repositories, and migration of nonaqueous-phase liquid pollutants in groundwater.