Stochastic modelling of contaminant transport in porous media

Abstract

The stochastic model of solute transport describes the motion of fluid elements in a porous medium as stochastic variation of the fluid velocity, produced by pore wall impacts. This gives a stochastic differential equation for fluid displacements, and solute transport is obtained as the cumulative effect over all realisations of the stochastic path. It is shown that applying Dynkin’s equation to obtain expectation values over realisations, a time and space dependent probability density for fluid displacements can be calculated. A new method is presented to calculate the evolution of solute concentration profiles from such probability densities. Applied to transport in a constant fluid drift velocity, the conventional diffusive description of the advection-dispersion equation is regained. However, with changing drift velocity e.g. in a non-homogenous porous medium, new effects emerge. A study of kinematical dispersion produced by advection is used to show that there are intrinsically stochastic effects, characterised as interaction between microscopic stochastic fluctuations and macroscopic velocity changes. This is demonstrated by analytical solution of the stochastic model for a linearly changing drift velocity. More general velocity changes are investigated by developing theoretical tools for piecewise constant and piecewise linear drift velocities. First, transmission of gaussian solute plumes through step changes in the drift velocity is shown to produce non-diffusive dispersion comparable to that found for linearly changing velocities. Approximations to the step results are developed that are simple enough to be applied for multiple steps allowing study of both a velocity staircase and velocity fluctuations about a constant average. It is shown that fluctuations enhance dispersion, and that the cumulative effect of a long series of fluctuations gives rise to a complex behaviour of dispersion with the traversal length through the porous medium, in accordance with observed behaviour commonly referred to as scale-dependent dispersivity. A particularly significant concept arising from the work is that of a natural length scale associated with velocity fluctuations, and which separates distinct short and long range dispersivity trends. To obtain full quantitative agreement with observed scale dependence over 5 orders of magnitude, the fluctuation model is extended by preserving its basic structure but taking the amplitude of the dispersion enhancement as variable. The resulting model has only a small number of parameters, largely constrained by plausibility considerations so that only one parameter is freely optimised, and explains the underlying mechanisms of the observed non-diffusive growth of dispersivity with traversal length. This explanatory aspect hinges to a large extent on the emphasis placed throughout on formulating analytical models rather than numeric ones, supported by the use of symbolic algebra software.