A spatial Markov model for upscaling transport of adsorbing-desorbing solutes

The Spatial Markov Model (SMM) is an upscaled model with a strong track record in predicting upscaled behavior of conservative solute transport across hydrologic systems. Here we propose an SMM that can account for reactive linear adsorption and desorption processes and test it on a simple benchmark problem: flow and transport through an idealized periodic wavy channel. The methodology is built using trajectories that are obtained from a single high resolution random walk simulation of conservative transport across one periodic element. Our approach encodes information about where a particle starts at the inlet, where it leaves at the outlet, how long it takes to cross the domain and one additional piece of information, the number of times a particle strikes the boundary, with the objective of predicting large scale transport with arbitrary linear adsorption and desorption rates. Our benchmark problem demonstrates that predictions made with our proposed SMM agree favorably with results from direct numerical simulations, which resolve the full transport problem.