Spatial distribution of solute particles in longitudinal dispersion due to advection

Previously we reported how to obtain a distribution of solute arrival times, W(t) due to advection. This calculation involved using cluster statistics of percolation theory in the framework of critical path analysis to find the likelihood that a given system size is characterized by a particular controlling conductance value, g, i.e., W(g,x), or W(g) for any given value of x. Then we calculated deterministically the scaling of the arrival time of particles through any particular pore size distribution along one of the tortuous paths characterized by W(g,x). Then W(t) was obtained from W(g)/(dt/dg), including known scaling properties related to the mass fractal dimensionality of the backbone percolation cluster. We now use this framework to generate a self- consistent expression for W(g,t) that a particle is traveling along a cluster with minimum g and then divide by dx/dg to obtain the distribution of particle distances at any time. While the former result had approximate power law tails in time, the present result resembles more closely a "stretched exponential" in space.