A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.