Fluid flow and solute transport in an anisotropic, heterogeneous porous medium with mean flow normal to a constant-flux boundary are considered. The statistical moments of flow and transport variables are determined at second order in log conductivity fluctuation, and they are expressed in terms of the log conductivity variance and integral scales, the mean flow velocity, and the distance from the boundary. The variance of the longitudinal and transverse components of velocity as well as hydraulic head variance and the longitudinal macrodispersivity are analyzed for a bounded medium with axisymmetric, Gaussian log conductivity covariance structure. In this case, all of the moments can be solved by means of a single numerical quadrature. The constant flow boundary increases the variability of head and of flow transverse to the mean flow direction and causes a reduction of the macrodispersivity in a zone adjacent to the boundary. Our results should be useful for the design and testing of numerical models and have important implications for surface infiltration of solutes. ¿ 2000 American Geophysical Union