The spatial, statistical structure of the fluid velocity field in the case of uniformly recharged heterogeneous aquifers is investigated, and the spatial covariances of the velocity field are derived. This information is necessary for an investigation of flow and macrodispersion using the Lagrangian formalism proposed by Dagan (1984). The resulting first two moments of the velocity are nonstationary. They are functions of a parameter &bgr; which characterizes the degree of flow nonuniformity and is related to the recharge. The displacement variances are computed and tested favorably using numerical simulations. Simple relations are developed which relate the transport parameters found for the case of uniform-in-the-average flows to nonuniform flows using a simple, nonlinear transformation of the travel time, based on &bgr;.