Scale-Dependent Porous Dispersion Resulting from the Cumulative Effects of Velocity Fluctuations

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.