  This work presents a hierarchical model for solute transport in bounded
layered porous media with random permeability. The model generalizes the
Taylor-Aris dispersion theory to stochastic transport in random layered porous
media with a known velocity covariance function. In the hierarchical model, we
represent (random) concentration in terms of its cross-sectional average and a
variation function. We derive a one-dimensional stochastic
advection-dispersion-type equation for the average concentration and a
stochastic Poisson equation for the variation function, as well as expressions
for the effective velocity and dispersion coefficient. We observe that velocity
fluctuations enhance dispersion in a non-monotonic fashion: the dispersion
initially increases with correlation length {\lambda}, reaches a maximum, and
decreases to zero at infinity. Maximum enhancement can be obtained at the
correlation length about 0.25 the size of the porous media perpendicular to
flow.
