Pulsed magnetic field gradient stimulated echo NMR measurements are reported for the flow and diffusion of an aqueous phase within a sample of Fontainebleau sandstone. The stimulated echo dependence on the gradient pulse area, q, for a fixed observation time Δ, is used to derive the displacement probability distributions, P Δ(X), where X, the direction of the field gradient, is arranged to be parallel or perpendicular to the applied pressure gradient driving the flow. The shape of the P Δ(X) as a function of Δ is modelled in terms of laminar flow within a set of randomly oriented capillaries. For small Δ, molecules remain largely within the capillaries they occupy at the beginning of the time Δ and, as a consequence, the P Δ(X) for small Δ are dominated by the orientational distribution of the capillaries. For large Δ, the evolution of the P Δ(X) is reproduced only when diffusion of molecules across the velocity gradients within the capillaries is introduced and, in addition, the fluid transfers between d...