Conventional and spatially resolved proton NMR and relaxation measurements are used in order to study the molecular motions and the equilibrium and nonequilibrium diffusion of oils in Berea sandstone and Venezuelan reservoir rocks. In the water-saturated Berea a single line with T*2 congruent to 150 microseconds is observed, while the relaxation recovery is multiexponential. In an oil reservoir rock (Ful 13) a single narrow line is present while a distribution of relaxation rates is evidenced from the recovery plots. On the contrary, in the Ful 7 sample (extracted at a deeper depth in a different zone) two NMR components are present, with 3.5 and 30 KHz linewidths, and the recovery plot exhibits biexponential law. No echo signal could be reconstructed in the oil reservoir rocks. These findings can be related to the effects in the micropores, where motions at very low frequency can occur in a thin layer. From a comparison of the diffusion constant in water-saturated Berea, D congruent to 5*10(-6) cm2/sec, with the ones in model systems, the average size of the pores is estimated around 40 A. The density profiles at the equilibrium show uniform distribution of oils or of water, and the relaxation rates appear independent from the selected slice. The nonequilibrium diffusion was studied as a function of time in a Berea cylinder with z axis along H0, starting from a thin layer of oil at the base, and detecting the spin density profiles d(z,t) with slice-selection techniques. Simultaneously, the values of T1's were measured locally, and the distribution of the relaxation rates was observed to be present in any slice.(ABSTRACT TRUNCATED AT 250 WORDS)