On the late‐time behavior of tracer test breakthrough curves

We investigated the late‐time (asymptotic) behavior of tracer test breakthrough curves (BTCs) with rate‐limited mass transfer (e.g., in dual‐porosity or multiporosity systems) and found that the late‐time concentration c is given by the simple expression c = t ad { c 0 g − [ m 0 (∂ g /∂ t )]}, for t ≫ t ad and t α ≫ t ad , where t ad is the advection time, c 0 is the initial concentration in the medium, m 0 is the zeroth moment of the injection pulse, and t α is the mean residence time in the immobile domain (i.e., the characteristic mass transfer time). The function g is proportional to the residence time distribution in the immobile domain; we tabulate g for many geometries, including several distributed (multirate) models of mass transfer. Using this expression, we examine the behavior of late‐time concentration for a number of mass transfer models. One key result is that if rate‐limited mass transfer causes the BTC to behave as a power law at late time (i.e., C ∼ t −k ), then the underlying density function of rate coefficients must also be a power law with the form α k−3 as a α → 0. This is true 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 residence time longer than the experiment, and possibly an infinite residence time, and also suggest an effective rate coefficient that is either undefined or changes as a function of observation time. We apply our analysis to breakthrough curves from single‐well injection‐withdrawal tests at the Waste Isolation Pilot Plant, New Mexico.