5.1 Mixed Convection Ratio

As noted earlier, many if not most convective groundwater systems are mixed convection systems where forced convection (flow driven by pressure or hydraulic head gradients) and free convection (flow driven by density variations) co-exist. They work together to control the flow behavior and solute concentration (or heat) distributions. In such cases, it is often interesting and important to understand the relative strengths of free and forced convection in controlling the resultant flow and transport process.

The mixed convection ratio is a simple and effective way to understand this. It is the ratio of the density-driven convective flow velocity to the advective flow velocity and allows determination of the dominant driving mechanism. The simplest form of a mixed convection ratio is for vertical flow and can be derived from Equation 18, which shows that the magnitude of the vertical flow is proportional to the sum of the vertical freshwater head gradient and the relative density difference (Equation 23).

(23)

In practical applications, the gradient term is replaced by the difference form (Equation 19), which approximates the gradient by taking the freshwater head difference, Δh_f, between two points separated by a distance Δz. Recall that density ρ in Equation 23 is the mean density between the two measurement points. The ratio of the two is the mixed convection ratio M (Equation 24).

(24)

where:

For the example case of stagnant saline groundwater considered earlier , so M = 1. For density-invariant flow, M = 0. If M >> 1, then free convection is dominant. If M << 1, then forced convection is dominant. For the example shown in Figure 7,

The definition of M according to Equation 24 applies to the analysis of vertical flow, but an equivalent mixed convection ratio can be defined to study instabilities in predominantly horizontal flow fields as well (Oostrom et al., 1992) as shown in Equation 25.

(25)

This form of the mixed convection ratio applies to the landfill example in Figure 12. Even though q_x varies spatially because there is groundwater recharge across the top of the aquifer (i.e., q_x increases from left to right), M’ can still be estimated if some effective q_x can be defined. In this case, it could be set to the specific discharge across the midpoint of the horizontal aquifer distance (x = 250 m). The volumetric flow rate due to the recharge (which is 0.5 mm d⁻¹) between x = 0 and x = 250 m is 5×10⁻⁴ m d⁻¹ × 250 m = 0.125 m² d⁻¹, which needs to be augmented by the inflow rate across the left boundary which was 0.5 m²d⁻¹ for this example. Assuming that the horizontal flow is distributed equally over the thickness of the aquifer gives q_x = (0.5 + 0.125) ⁄ 40 = 0.016 m d⁻¹. The hydraulic conductivity was K = 25 m d⁻¹, and equates to 3.25 × 10⁻⁴, 3.25 × 10⁻³ and 1.3 × 10⁻², for the low, medium, and high-density cases, respectively. Inserting these values into Equation 25 yields corresponding values for M’ of 0.52, 5.2 and 20.8. The values are all above the critical threshold of M’ ≈ 0.3 defined by Oostrom and others (1992), so one would expect the plume to be unstable for all density contrasts considered, yet the low-density case in Figure 12 is stable. One likely reason for this is that there is recharge across the top boundary, which is different from the experimental conditions in Oostrom and others (1992). Moreover, the Oostrom and others (1992) criterion was based on laboratory experiments, in which the physical heterogeneities that trigger instabilities are much more pronounced than in the numerical simulation of the landfill leachate plume, in which small numerical roundoff errors form the triggering instabilities. This aspect is discussed in more detail in Section 6.

Mixed convection ratios have been used by various authors to assess the relative importance of free and forced convection in regional flow systems. For example, Illangasekare and others (2006) used it to infer the main driving force of aquifer salinization in Sri Lanka following the December 26, 2004 tsunami, while Holzbecher (2005) used it to determine the effect of regional flow on the flow pattern and salinity distribution near salt lakes. Mixed convection ratios can be formulated for more complex flow conditions as well. For example, Ward and others (2009) give expressions for M for radial flow fields.