Teaching and communicating dispersion in hydrogeology, with emphasis on the applicability of the Fickian model

Abstract

The process of dispersion in porous media is the effect of combined variability in fluid velocity and concentration at scales smaller than the ones resolved that contributes to spreading and mixing. It is usually introduced in textbooks and taught in classes through the Fick-Scheidegger parameterization, which is introduced as a scientific law of universal validity. This parameterization is based on observations in bench-scale laboratory experiments using homogeneous media. Fickian means that dispersive flux is proportional to the gradient of the resolved concentration while the Scheidegger parameterization is a particular way to compute the dispersion coefficients. The unresolved scales are thus associated with the pore-grain geometry that is ignored when the composite pore-grain medium is replaced by a homogeneous continuum. However, the challenge faced in practice is how to account for dispersion in numerical models that discretize the domain into blocks, often cubic meters in size, that contain multiple geologic facies. Although the Fick-Scheidegger parameterization is by far the one most commonly used, its validity has been questioned. This work presents a method of teaching dispersion that emphasizes the physical basis of dispersion and highlights the conditions under which a Fickian dispersion model is justified. In particular, we show that Fickian dispersion has a solid physical basis provided that an equilibrium condition is met. The issue of the Scheidegger parameterization is more complex but it is shown that the approximation that the dispersion coefficients should scale linearly with the mean velocity is often reasonable, at least as a practical approximation, but may not necessarily be always appropriate. Generally in Hydrogeology, the Scheidegger feature of constant dispersivity is considered as a physical law and inseparable from the Fickian model, but both perceptions are wrong. We also explain why Fickian dispersion fails under certain conditions, such as dispersion inside and directly upstream of a contaminant source. Other issues discussed are the relevance of column tests and confusion regarding the meaning of terms dispersion and Fickian.