Spatial Markov Model of Advective‐Diffusive Transport in Heterogeneous Domains

Abstract Spatial Markov models (SMM) are an efficient approach to simulate transport in heterogeneous media across scales. They represent particle transport by equidistant spatial transitions with correlated random velocities, which renders the associated transition times correlated random variables. While SMM perform excellently for modeling purely advective transport, incorporating diffusion remains challenging. So far, applying SMM to advective‐diffusive transport in porous media has been mostly restricted to using empirical transition matrices based on numerical simulations. Using a transition matrix for advective‐diffusive transport obliterates the fundamental differences between the two transport processes and is conflicting with the goal of replacing explicit transport simulations by a SMM. Here, we present an advective‐diffusive SMM that conceptualizes diffusion as jumps between advective trajectories, that is, diffusion competes with advection for changing the particle velocity. At each particle transition, a random diffusion time is compared to the current advection time. If the advection time is shorter than the diffusion time, the particle remains on its current SMM trajectory and the longitudinal velocity correlation is kept. If the diffusion time is shorter, the particle velocity is reset. Breakthrough curves and their first and second moments calculated with the advective‐diffusive SMM are in agreement with three‐dimensional numerical simulations in heterogeneous log‐conductivity fields with isotropic, exponential covariance function with variances up to five.