Slow advection and diffusion through low permeability inclusions

Heterogeneous porous media with low permeability lenses can produce contaminant plumes with extended tails. Highly asymmetric breakthrough curves cannot be described well by an advection–dispersion equation (ADE) with a uniform velocity and dispersion coefficient. The character of such a solute plume depends on many factors, including plume size, the geometry and arrangement of the low permeability inclusions, and the transport through such regions. We develop an inclusion Peclet number that effectively characterizes the relative importance of advection and diffusion for transport within a low permeability lens. This inclusion Peclet number is a function of the far-field velocity, the effective diffusion coefficient, the length scale of the inclusion, and the ratio of the permeability of the lens to that of the surrounding matrix. We investigate the effects of a single, elliptical inclusion on the arrival of a solute at a downgradient control plane with numerical particle tracking. Effects specific to advection or diffusion dominance within the inclusion are subtle: diffusion gives rise to more distributed tailing whereas advection produces behavior that is more abrupt. These slight differences are not enough to allow one to determine the dominant process within the inclusion by observing the first three temporal moments alone. The time scale for the dominant transport process within the inclusion is the primary factor affecting the contaminant tailing. For high inclusion Peclet numbers (when advection dominates), the characteristic time varies with the permeability contrast, the far-field velocity, and the size and geometry of the inclusion. For low inclusion Peclet numbers (when diffusion dominates), the characteristic time varies with the size of the inclusion and the effective diffusion coefficient.