We examine the utility of the lattice Boltzmann method for modeling fluid flow in large microstructures. First, results of permeability calculations are compared to predicted values for several idealized geometries. Large scale simulations of fluid flow through digitized images of Fontainebleau sandstone, generated by X-ray microtomography, were then carried out. Reasonably good agreement was found when compared to experimentally determined values of permeability for similar rocks. We also calculate relative permeability curves as a function of fluid saturation and driving force. The Onsager relation, which equates off-diagonal components of the permeability tensor for two phase flow, is shown not to hold for intermediate to low nonwetting saturation, since the response of the fluid flow to an applied body force was nonlinear. Values of permeability from three phase flows are compared to corresponding two phase values. Performance on several computing platforms is given.