Time-Dependent Dispersion of Nonergodic Plumes in Two-Dimensional Heterogeneous Aquifers

Abstract

The time-dependent second spatial moment Z(t) ≡<S(t)>−S(0) and the effective dispersivity γ(t), defined as (1/2μ)[d<S>/dt], of nonergodic plumes are evaluated in two-dimensional (2D) statistically isotropic media under a uniform mean velocity μ, and the first-order approximation of the particle displacement, to study the effect of initial plume size on Z(t) and γ(t), where <S(t)> is the ensemble average of the second spatial moments, S(t), of a solute plume about its center of mass, and S(0) is the initial value of S(t). It is found that the aspect ratio ι2/ι1 of an area source plays an important role in the dispersive behavior of a nonergodic plume, where ι2 and ι1 are the normalized lengths (with respect to the correlation scale of log-hydraulic conductivity) of the source normal and parallel to μ, respectively. When ι2/ι1≥ 1, both the longitudinal and transverse second spatial moments Z11 and Z22 increase at early time, and at large time Z11 becomes proportional to time, or the effective longitudinal dispersivity γ11 approaches a constant while Z22 becomes constant or γ11 approaches zero. When ι2/ι1< 1, Zii and γii (i= 1, 2) increase at early time. At late time, however, they behave in a fundamentally different manner: Z11 increases monotonically to approach a constant, or γ11 decreases gradually to zero while Z22 reaches a peak and then decreases to zero, or γ22 falls below zero, i.e., becomes negative, and increases again to approach zero. Comparison of the first-order theoretical results with a Monte Carlo simulation shows good agreements for plumes with large initial sizes in less heterogeneous media.