# 5 Dimensionless Numbers

Unstable density layering may not always give rise to convective flow. It depends on several factors. Instability tends to occur where the permeability of the geologic material is high (as in sand) or where density differences are large (e.g., beneath an evaporating salty lake). Diffusion (for solutes) or conduction (for heat) also play a role. Free convection will occur when small perturbations of the concentration or temperatures can grow into larger finger-like structures. Over time, diffusion and conduction will tend to suppress the formation of instabilities as these processes smear out concentration and temperature differences, and the onset of fingering thus depends on their ability to suppress the growth of perturbations. But how can we predict whether a system is stable or unstable, and what conditions determine this? Also, how can we determine the importance of density-driven flow relative to natural background flow in a mixed convection system?

One useful way to assess the importance of density driven flow in a groundwater system is to test the sensitivity of flow and transport behavior in a numerical groundwater model. One can run the model with the density coupling turned on so that as the concentration or temperature field changes, the fluid density changes. This is called a density-variant solution or density-variant model. The density-variant model can be compared with the density-invariant model, in which concentration or temperature changes do not affect the density. If the results of the two cases are very similar or identical, one can conclude that density is likely an insignificant part of the system behavior. If there is a significant difference between the two cases, then one can conclude that density is an important factor and that this may warrant further investigation and inclusion in subsequent analyses. Section 6 discusses variable-density groundwater models in more detail.

Not all numerical codes have the option to consider density effects, and variable-density groundwater models are much more complex than constant-density models. Setting up a numerical model takes time and is not always easy nor trivial. Alternatively, dimensionless numbers can be used to assess the importance of density variations and serve as a useful starting point before more complex analyses are undertaken. Theoretical treatments are given in books such as Holzbecher (1998) and Nield and Bejan (2006). Only salient details are described in the following subsections.