A travel time approach to exclusion on transport in porous media

Abstract

Transport of suspended colloidal particles in porous media often appears faster relative to molecular‐scale tracers under the same flow field in both laboratory and field studies. A similar phenomenon appears in the transport of ionic molecules in like‐charged fine‐grained porous media and is known as anion exclusion. Here a simplified empirical model of the travel time density function for excluded tracer arrivals is developed in terms of the travel time density of an unexcluded tracer in the same flow field by equating the excluded particle trajectories with unexcluded particle trajectories modified with “detours.” The detours are defined generally without invoking assumptions about the detoured trajectory velocities or proximity to pore walls, although such may be introduced. This conceptual basis provides a representation of the effects of exclusion on the transport alone, exclusive of reactions terms, that is consistent with respect to mass balance as well as particle kinematics. The analysis involves a constitutive “speedup” function that tells how travel times of unexcluded particles map to those of excluded particles. A simple inverse operator that identifies the speedup function given experimentally observed cumulative arrival distributions (e.g., breakthrough curves) of excluded and unexcluded particles in the same flow field is derived. The inverse analysis requires breakthrough curve data for which the reactions and transport are separable, as in the case of irreversible kinetic reactions. The relations developed are exercised in calculating the breakthrough curve of excluded particles given the breakthrough curve of a neutral and inert molecular tracer. The method is then applied to recently published data to determine an empirical speedup function, which is found to be in this case invariant of solute flow rate.