Nonergodic solute transport in three-dimensional heterogeneous isotropic aquifers

The general expressions for the time-dependent ensemble averages of the second spatial moments ^A& and the effective dispersivities g, defined as (1/ 2m)(d^A&/dt) where m is the magnitude of the mean flow velocity m, are evaluated in order to study the effect of initial plume size on ^A& and g in three-dimensional heterogeneous isotropic aquifers under the first-order approximation to the particle displacement. The results confirm previous findings that ^A& and g generally approach their respective ergodic limits X and a as the size of a source increases, where X and a are the single particle displacement covariance and the associated dispersivity, and that the transverse lengths of a source are more important than the longitudinal length for the ergodic condition to be met. The longitudinal dispersion of a nonergodic plume becomes Fickian or the effective asymptotic longitudinal dispersivity is constant at late time as long as one of the initial lateral lengths of the plume is nonzero, while the transverse dispersion is always nonFickian and the effective asymptotic transverse dispersivities are always zero regardless of the initial plume size. The most important and interesting findings are, when the longitudinal length l1 of an initial plume is larger than the lateral lengths l2 and l3, both effective longitudinal and transverse dispersivities g ii (i 5 1, 2, 3) increase to their respective peaks at early time, then g11 decreases toward an asymptotic constant, whose value depends on the values of l2 and l3 (g11 3 0 if l2 5 l3 5 0), whereas g22 and g33 decrease to below zero (i.e., become negative), increase again, and finally approach zero independent of the lateral lengths of the source. Comparison of the current study with a numerical simulation shows good agreement between the calculated and simulated longitudinal second spatial moments.