On the late-time behavior of tracer test breakthrough curves

We investigated the late-time (asymptotic) behavior of tracer test breakthr ough curves (BTCs) with rate-limited mass transfer (e.g., in dual-porosity or multiporosity systems) and found that the late-time concentration c is g iven by the simple expression c = t(ad){(c(0)g - [m(0)(partial derivativeg/ partial derivativet)]}, for t much greater than t(ad) and t(alpha) much gre ater than t(ad), where t(ad) is the advection time, co is the initial conce ntration in the medium, mo is the zeroth moment of the injection pulse, and t, is the mean residence time in the immobile domain (i.e., the characteri stic mass transfer time). The function g is proportional to the residence t ime distribution in the immobile domain; we tabulate g for many geometries, including several distributed (multirate) models of mass transfer. Using t his expression, we examine the behavior of late-time concentration for a nu mber of mass transfer models. One key result is that if rate-limited mass t ransfer causes the ETC to behave as a power law at late time (i.e., c simil ar to t(-k)), then the underlying density function of rate coefficients mus t also be a power law with the form alpha (k-3) as alpha --> 0. This is tru e for both density functions of first-order and diffusion rate coefficients . BTCs with k < 3 persisting to the end of the experiment indicate a mean r esidence time longer than the experiment, and possibly an infinite residenc e time, and also suggest an effective rate coefficient that is either undef ined or changes as a function of observation time. We apply our analysis to breakthrough curves from single-well injection-withdrawal tests at the Was te Isolation Pilot Plant, New Mexico.