Effective ADE models for first-order mobile-immobile solute transport: Limits on validity and modeling implications

Quasi-1D mobile-immobile transport processes which have exponentially distributed random waiting times in both mobile and immobile states are common in hydrologic models (for example, of transport subject to kinetic sorption). The central limit theorem implies that eventually such transport will be expressible with an effective ADE (i.e. a generalization of the common retardation factor approach with an added Fickian dispersion coefficient accounting for the effect of trapping). Previous works have determined formulae for the value of this coefficient based on the transport properties. However, the time until convergence to Gaussian behavior has not previously been quantified. To this end, exact Green's functions characterizing the transport at all times are derived for the case of pure advection. The Green's functions are expressed in terms of three dimensionless parameters, representing location, time, and capacity coefficient. In the pre-Gaussian regime, a parametric study characterizing concentration profile asymmetry as a function of the capacity coefficient is performed. Next, heuristics are presented in terms of the dimensionless parameters for the time until the effective ADE adequately reflects reality. For strongly retarded solute, the time until effective ADE validity is found inversely proportional to release (e.g., desorption) rate. The nature of the effective dispersion coefficient is examined, and the possibility of large trapping-driven dispersion even in cases where batch experiments would detect negligible trapping is demonstrated. Collectively, these results call into question reliance on retardation factors derived from batch experiments for many practical transport modeling efforts; knowledge of both the trapping and release kinetics appears essential.