4.1 Free Convection

The potentially substantial effect of density on groundwater flow is illustrated by the following experiment, which was conducted by Simmons and others (2002b) to study the impacts of unstable flow in a laboratory sand tank and is shown in Figure 8. The tank has dimensions of about 1.2 m by 1.2 m and is filled with homogeneous sand, saturated with fresh water at time t = 0. A calcium chloride plume of 3000 mg L⁻¹ was introduced into the top of the sand tank. The plume was stained purple with a dye to allow visualization. Figure 8a shows the resulting plume at 150 minutes at which time the plume has moved fairly uniformly through the sand to about 25 to 30 cm depth. The process was repeated with a much higher concentration fluid introduced to the top of the tank. This time the calcium chloride solution had a concentration of 300,000 mg L⁻¹, so one hundred times more than the first low concentration plume. Figure 8b shows the resultant plume at 50 minutes (only one-third of the time of the result shown for the low concentration plume in Figure 8a). The difference between the results is striking and remarkable. In the case of the higher density plume, the plume displays substantial instability with dense lobes sinking in the tank under the influence of gravity. This process is often referred to as fingering due to the resemblance of the lobes with fingers. In between each sinking lobe, fresher water is upwelling. In just one-third of the time, the high-salinity plume has moved about three times the distance traversed by the low-salinity plume.

Figure 8 – a) Plume in a sand tank: 3 g L^–1 CaCl₂ after 150 min since the start of the experiment. b) Plume in a sand tank: 300 g L^–1 CaCl₂ after 50 min since the start of the experiment. (From Simmons et al., 2002b). A video of the experiment is available here.

This experiment illustrates the effect of density-driven flow simply and beautifully. It shows the development of gravitational instabilities and the profound implications of such instabilities for flow and transport, which are:

• an increase in the total quantity of solute involved in the solute transport process;
• a significant reduction of the time scales for mixing over similar spatial scales; and
• an increase in the distance of solute spreading.

In short, unstable, transient, density-driven flow is a rapid and effective spreading and mixing mechanism for solutes.

The flow shown in Figure 8b is driven purely by density variations and is called free (or natural) convection. The large density contrast in the experiment exacerbates the role of density. It would be a misconception though to associate density-driven flow with only high solute concentrations. Even low-density differences can exert a significant effect on the flow pattern. This was made clear in the famous Cape Cod tracer test, which involved the injection of a mixture of Li⁺ salts (LiBr, LiF and Li₂MoO₄) dissolved in water into a sand and gravel aquifer. The total concentration of dissolved solids was approximately 890 mg L^–1 and the density difference ∆ρ of the injected solution with the ambient groundwater varied between 0.74 and 0.85 kg m^–3 (LeBlanc et al., 1991).

Numerical modeling confirmed that the density difference, albeit small, was sufficient to cause the injected plume to sink (at a rate higher than what would be expected due to rainfall recharge) during the early stage of the injection experiment (Zhang et al., 1998). During later stages, density-driven flow decreased in importance as the plume mixed with the ambient groundwater.

The downward flow rate associated with convective fingering tends to be much higher relative to the vertical flow rates in groundwater systems driven by topography and rainfall recharge. The speed of finger descent (v_z, m s⁻¹) in systems without an ambient flow field was given by Wooding (1969) as Equation 20.

(20)

where:

For an aquifer with k = 10⁻¹¹ m² and porosity 0.4 which is in the range of property values for a clean sand, this means that v_z ≈ 2.5 × Δρ when expressed in m y^–1. So, if seawater floods a freshwater aquifer, salt fingers may be expected to sink at a rate of approximately 2.5 × 25 = 62 m y⁻¹. Higher rates apply below sabkhas (flat areas that are regularly inundated where salts precipitate when water evaporates). For example, van Dam and others (2009) observed salt fingers that moved down 15 m in a seven-week period. It is interesting to compare such vertical flow velocities to those generated by groundwater recharge, which are on the order of millimeters per year in arid- and semi-arid regions, to typically one or two meters per year in humid regions. The speed of fingering in groundwater is a remarkable observation that defies conventional wisdom about the rate of groundwater processes. In more typical advection-driven situations, much smaller flow rates are usually encountered. In relative terms, fingering associated with free convection can be a remarkably fast groundwater process. Convective fingering, where it occurs, can completely alter the solute distribution in a short period of time.