Linear Stochastic Analysis of the Partial Reversibility of Ensemble and Effective Dispersion in Heterogeneous Porous Media

Macrodispersion in heterogeneous formations is caused by spatial variability of the velocity field, making different parts of a plume experience different advective displacements. Differential advection interacts with diffusion, which hardly affects longitudinal ensemble dispersion (i.e., the spread of the ensemble-averaged concentration) but determines effective dispersion, that is, the expected spread of individual plumes for point injections. The latter has been suggested as metric of solute mixing. Pure advection is fully reversible, whereas diffusion is completely irreversible. We quantify the partial reversibility of macrodispersion by analyzing the second central ensemble and effective spatial moments for advective-diffusive transport in heterogeneous domains with flow reversal, applying linear stochastic theory to approximate the corresponding moments and comparing them to particle-tracking random-walk simulations in periodic media. Diffusion causes solute particles to deviate from their forward trajectories when flow is reversed. As long as advective memory dominates, both types of second central moments decrease during backward motion, then reach a minimum, and increase again. The reversibility is considerably bigger for ensemble than effective dispersion but the latter also shows partial reversibility, challenging its use as metric of mixing. The stronger diffusion is in comparison to advection, the less reversible dispersion becomes. After equally long times of forward and backward motion, the two types of second central moments differ, but to a much smaller extent than in pure forward motion. In realistic settings, the advective memory affects dispersion so strongly that the asymptotic regime is not reached before the plume center has returned to its origin.