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 b.Delta, is used to derive the displacement probability distributions, P_b.Delta(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_b.Delta(X) as a function of b.Delta is modelled in terms of laminar flow within a set of randomly oriented capillaries. For small b.Delta, molecules remain largely within the capillaries they occupy at the beginning of the time b.Delta and, as a consequence, the P_b.Delta(X) for small b.Delta are dominated by the orientational distribution of the capillaries. For larger b.Delta, the evolution of the P_b.Delta(X) is reproduced only when diffusion of molecules across the velocity gradients within the capillaries is introduced and, in addition, the fluid transfers between differently oriented capillaries through the action of the overall flow.