On a new functional form for the dispersive flux in porous media

Abstract

A recently developed second‐order model for local dispersive transport in porous media has been simplified to yield a new, closed‐form relationship for the dispersive flux. In situations characterized by negligible velocity gradients, the flux can generally be represented as a convolution or “memory” integral over time of previous concentration gradients. The strength of this memory is controlled by an exponential weighting factor related to the magnitudes of the velocity and local molecular diffusive flux. The form of this result is consistent with other models of diffusive and dispersive transport phenomena over various spatial scales. In circumstances where the memory strength is small, the integral can be simplified and cast in the form of a standard Fickian relationship with apparent time‐dependent dispersivity functions that grow to finite, asymptotic values. This specific formulation can be manipulated to yield a one‐equation transport balance law in the form of a telegraph equation. Nonphysical effects, such as spurious upstream dispersion and instantaneous propagation of mass to extremely distant points predicted with a Fickian law, are reduced or eliminated. Although the importance of the new result in transport simulations will depend on the spatial and temporal scales of interest, it should provide some insight in the interpretation and design of new experiments.