Three‐dimensional stochastic analysis of macrodispersion in aquifers

TLDR

The theory shows that at large displacements a classical Fickian gradient transport relationship holds. The study analyzes dispersive mixing in three‑dimensional heterogeneous porous media using stochastic continuum theory. Stochastic solutions of perturbed steady‑flow and solute‑transport equations are used to build the macroscopic dispersive flux and compute the macrodispersivity tensor from a three‑dimensional, statistically anisotropic hydraulic‑conductivity covariance. For isotropic conductivity, longitudinal macrodispersivity is convectively controlled while transverse macrodispersivity is much smaller; for anisotropic conductivity all tensor components are convectively controlled with a transverse‑to‑longitudinal ratio of ~10⁻¹, significant off‑diagonal terms, and highly anisotropic transverse dispersion, all predictions agreeing with field experiments and simulations.

Abstract

The dispersive mixing resulting from complex flow in three‐dimensionally heterogeneous porous media is analyzed using stochastic continuum theory. Stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three‐dimensional, statistically anisotropic input covariance describing the hydraulic conductivity. With a statistically isotropic input covariance, the longitudinal macrodispersivity is convectively controlled, but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term. With an arbitrarily oriented anisotropic conductivity covariance, all components of the macrodispersivity tensor are convectively controlled, and the ratio of transverse to longitudinal dispersivity is of the order of 10 −1 . In this case the off‐diagonal components of the dispersivity tensor are significant, being numerically larger than the diagonal transverse terms, and the transverse dispersion process can be highly anisotropic. Dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and Monte Carlo simulations. The theory, which treats the asymptotic condition of large displacement, indicates that a classical gradient transport (Fickian) relationship is valid for large‐scale displacements.