Temporal moments for transport with mass transfer described by an arbitrary memory function in heterogeneous media

Abstract

Temporal moment equations are generalized for transport under linear mass transfer, which has been used to model a broad range of small‐scale processes: kinetic sorption, diffusion into immobile regions, and transport through heterogeneous aquifers. Solving the moment equations, which are formally identical to steady state transport equations, is computationally more efficient than evaluating the temporal moments by integrating the transient flux concentrations. We derive recursive relations for the moments of the flux concentration, which involve the moments of the memory function but do not dependent on its shape. It turns out that two mass transfer models have the same kth temporal moment if the moments of order lower than k are equal. Particularly, the mean retention time, i.e., the first moment of the retention probability density function (pdf) in the immobile domain, decides the second temporal moment of concentration. A mass transfer model with two first‐order rate coefficients can match up to the fourth temporal moment described by a multirate model with a predescribed pdf of the mass transfer rate coefficient. The kth temporal moment is finite when the (k‐1)th moment of the memory function exists.