Pulsed-held-gradient NMR measurements of the one-dimensional probability-density distribution P Delta (X) for diffusive displacements of water molecules in times Delta in a water-wet sandstone, saturated with a mixture of dodecane and brine, are presented for water saturations S-w less than or equal to 0.32. The distributions show a distinctive shape which is attributed to the distribution of the aqueous phase in thin, locally curved and effectively two-dimensional layers on the pore surface. The data are reproduced well by a simple model in which diffusion is restricted to the surfaces of an isotropic array of infinite planes, with an effective diffusion coefficient D-2(Delta) which shows a systematic dependence on Delta. This is suggested to arise, in the main, from local curvature of the pore surface.