5.3 Wooding Number

Vincent E.A. Post; Craig T. Simmons

5.3 Wooding Number

As mentioned in the previous section, application of the Rayleigh number is limited to systems without a background flow field. Wooding (1960) analyzed the case of a boundary layer that grows by diffusion in the presence of a flow that opposes diffusion. Situations like these are found below the bottom of salt lakes that receive groundwater input by upward seepage (Simmons et al., 2002; Simmons et al., 1999; Wooding et al., 1997), salt pans (Bauer et al., 2006) or where groundwater discharges into the ocean (Greskowiak, 2014). A stable boundary layer occurs when the diffusive flux equals the opposing advective flux (Figure 15).

Schematic representation of a flow system

Figure 15 – Schematic representation of a system with a flow component (upward in this image) in the opposite direction as diffusion (downward in this image), where a density difference ∆ρ exists over the characteristic boundary layer thickness δ. The hatched area is a zone of constant concentration that is higher than the initial concentration in the porous medium. The curved line represents the fluid density distribution below the hatched zone.

Wooding et al. (1997) defined the characteristic thickness δ of the boundary layer as shown in Equation 28.

$\displaystyle \delta =\frac{D}{q_{z}}$

(28)

When H is replaced by δ in Equation 26, the boundary layer Rayleigh number, or Wooding number, becomes Equation 29.

$\displaystyle Ra^{\delta }=\frac{\Delta \rho gk\delta }{\mu D}=\frac{\Delta \rho gk}{\mu q_{z}}$

(29)

Based on linear perturbation analysis, Wooding (1960) determined the value of the critical boundary Rayleigh number to be $Ra_{cr}^{\delta }\approx 7$ . For values less than 7 which are situations with high q_z or low Δρ, the boundary layer initially increases and then reaches a stable thickness. For values greater than 7 which are situations with low q_z or high Δρ, the boundary layer continues to increase and never stabilizes.

High values of q_z tend to stabilize the system, as shown in Figure 16. The colored lines mark the boundary between the stable and unstable regime for a range of values of q_z that may be encountered in groundwater systems. Fluxes of q_z > 0.1 m d⁻¹ are required to prevent instabilities from amplifying in the most permeable aquifers.

Graph showing stable and unstable regimes

Figure 16 – Boundaries between stable and unstable regimes for different values of q_z (in m d⁻¹), indicated by colored lines. The lines are calculated according to Equation 29 with $Ra=Ra_{cr}^{\delta }=7$ . Stable conditions exist to the left, and unstable to the right, of each line marking the boundary between the regimes. Values of q_z well above 0.1 m d^–¹ would be required to maintain stable conditions in gravels for even small density differences.

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