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W.S. Verwoerd
Applied Mathematical Modelling, Volume 35, Issue 7, July 2011
Abstract
The stochastic solute dispersion model studied in the previous article, can be applied to more realistic velocity variations by approximating them as piecewise constant. This requires treatment by a boundary value formulation, which raises problems connected with entropy considerations. A method is developed to deal with these by the introduction of a specially designed compensator function into the boundary value probability integral for calculating solute concentration. Applying this even for a single velocity step yields an intractable integration, but a suitable approximation is constructed that allows it to be evaluated in analytical form. The result is that a Gaussian solute plume impinging on a velocity step is transmitted as a modulated and compressed or dilated quasi-Gaussian. Plume dispersion is encapsulated in an enhancement factor F that multiplies the diffusive, linear time, dispersion. F is also time dependent; at the time of step penetration it equals kinematical dilation, but anneals away non-linearly so that a length scale can be established over which downstream effects of a velocity step on the dispersion extends.
With multiple steps, the cumulative effect is expressed by a product of enhancement factors, each related to a single step. While kinematical compression and dilation effects cancel out across each stepped fluctuation, stochastic dispersion exceeds diffusive values for any plausible combination of fluctuation amplitude and length. A simple formula is obtained for the effective enhancement factor of a fluctuation in terms of its length and amplitude. Moreover, the algebra leads to the definition of a natural length scale Λ related to the Peclet number of the flow. Algebraic evaluation of the cumulative dispersion enhancement by a sequence of equally spaced identical fluctuations, leads to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model in agreement with observations are that the dispersivity behaves differently for traversal lengths above and below Λ, and that above this transition it is proportional to a fractional power of the traversal length. Simple stepwise models do not produce the full extent of the observed exponential increase below the transition, but modifying F by introduction of an adjustable parameter addresses this and leads to good agreement with experimental values over a range of 5 orders of magnitude in both traversal length and dispersivity.
Additional Information:
“The fundamental mechanism underlying scale-dependent porous dispersion is presented. This is based on interpretation of a semi-analytical model of Gaussian solute plume transmission through a stepwise fluctuating drift velocity . A key observation is that a fluctuation enhances diffusive dispersion as shown in the figure, and the effects of a sequence of fluctuations accumulate as a product. This results in an exponential rise of dispersion with the distance travelled. However, this behaviour is tempered by an annealing effect downstream of each velocity step, giving rise to a traversal length scale Λ. The combined effects of the productwise accumulation of declining dispersion enhancements, is that dispersion at first rises exponentially, but when the traversal length approaches Λ there is a transition to a slower power law rise.”
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