Understanding the non-Gaussian nature of reactive solute transport. From   particle dynamics to the partial differential equations
  In the present study we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher's equation. Characteristic for this system is a longitudinal spreading, that becomes Gaussian only in the long-time limit. We refer to this as kinetics induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics induced non-Gaussian spreading. In the long run the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume this interdependence is displayed as a `tailing' effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion) we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case it appears that the transverse spreading depends on the longitudinal coordinate. This results in `cigar-shaped' contours. 
Gerard Uffink and Amro Elfeki and Michel Dekking and Johannes Bruining   and Cor Kraaikamp
math.PR cond-mat.stat-mech physics.flu-dyn