Dispersion is the process by which particles entrained in flow through a porous medium spread in time and space. This spreading is seldom Gaussian, having much fatter (power-law) tails, and cannot be predicted using conventional partial differential equations. We build on an existing procedure that utilized prior calculations of the distribution of particle velocities and path lengths to predict the distribution of solute arrival times neglecting molecular diffusion. The calculations were based on a fusion of percolation theoretical techniques: cluster statistics of percolation theory, critical path analysis, and path tortuosity. Each flow path is defined through the smallest (rate-limiting) conductance on that path. The long-time tail in the distribution of arrival times relates to those tortuous paths of particle transport near the percolation threshold. When diffusion is included, however, particles cannot stay on these paths for such long times and power law behavior grades into Gaussian behavior at long distances (as is already known to occur for an individual capillary). Effects of diffusion relative to flow are estimated using a quantity called the Peclet number, Pe. We find that Gaussian behavior sets on at a distance which is proportional to a power of Pe. Our proposed research includes: 1) Generating the probability, fi, that a particle diffuses off a flow path at pore i, 2) Using fi to generate the probability that a particle stays on a particular flow path to an arbitrary length and incorporate into existing code, 3) Relating the resulting distribution of arrival times, W(t), at arbitrary spatial position, x, to a spatial solute distribution, W(x) at arbitrary time, t, 4) Calculating the moments of the spatial solute distribution. Understanding spatio-temporal evolution of the concentrations of solutes is important in monitoring subsurface spills, evaluating groundwater age distributions, guiding plant uptake of minerals and fertilizers, and design of enhanced oil-recovery processes. Thus this research can enhance understanding of processes relevant to agriculture, oil and mining industries, toxic waste clean up and risk assessment, geological sequencing, and groundwater age and source estimation. The second potential impact of our work is that it could lead to a change in perspective of the people in this field of research, with renewed emphasis on techniques other than those of continuum mechanics and the associated differential equations.
