Solvable continuous time random walk model of the motion of tracer particles through porous media

Abstract

We consider the continuous time random walk model (CTRW) of tracer's motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported in Holzner et al. Phys. Rev. E 92, 013015 (2015). The particle's passing through one channel is modelled as one step of the walk. The step's (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step's duration τ, length l and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter α. Depending on that parameter all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds that was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer's diffusion in the porous medium is considered.