Abstract
The central proposition of this work is that microscopic scattering induced by pore walls on fluid flow transforms the fundamental equation of motion into a stochastic partial differential equation (SPDE), in which the driving coefficient, the velocity, has a stochastic component. Flow velocity variation on the macroscopic scale, as in an inhomogeneous aquifer, causes non-diffusive dispersion which is the root cause of the observed dependence of dispersivity on the length scale of the flow. For the case of a 1D constant velocity gradient, exact analytical solution shows that non-kinematic contributions do not cancel over the course of a triangular velocity fluctuation.
Approximate analytical modelling of the Gaussian plume transmission across a velocity step is followed by the case of a stepwise fluctuation, and hence a sequence of fluctuations. Combining steps to form a fluctuation causes a net increase of dispersion compared with simple diffusion. Multiple fluctuations combine productwise, giving rise to a natural length scale. This divides the dispersion growth into distinct ranges: an exponential low-range growth followed by power law growth in the high range. This agrees with the observed behaviour; plausible extensions preserving the algebraic structure give a full quantitative account of the scale-dependent dispersion phenomenon.
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Verwoerd, W. (2014). Scale-Dependent Porous Dispersion Resulting from the Cumulative Effects of Velocity Fluctuations. In: Basu, S., Kumar, N. (eds) Modelling and Simulation of Diffusive Processes. Simulation Foundations, Methods and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-05657-9_7
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DOI: https://doi.org/10.1007/978-3-319-05657-9_7
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