A note on the theoretical foundations of particle tracking methods in heterogeneous porous media

In this technical note the stochastic process for the path of particles undergoing convection dispersion in a heterogeneous media with sharp interfaces is derived. The derived process is shown to correspond to the numerical particle tracking scheme introduced on a more heuristic basis by Hoteit et al. (2002) who validated the accuracy of their numerical method in controlled laboratory experiments. Specifically, it is shown that in the direction transverse to the flow U(y) and across an interface where the diffusion coefficient is discontinuous the stochastic particle path t → Y*(t) is given by a rescaled version of Itô and McKean's (1963) α-skew Brownian motion for a uniquely determined value of α that depends on the values of the diffusion coefficient at either side of the interface. These results help supply a theoretical foundation that supports the numerical schemes and experimental findings of Hoteit et al. (2002).