On the late-time behavior of tracer test breakthrough curves

The authors investigated the late-time (asymptotic) behavior of tracer test breakthrough curves (BTCs) with rate-limited mass transfer (e.g., in dual or multi-porosity systems) and found that the late-time concentration, c, is given by the simple expression: c = t{sub ad} (c{sub 0}g {minus} m{sub 0}{partial_derivative}g/{partial_derivative}t), for t >> t{sub ad} and t{sub a} >> t{sub ad} where t{sub ad} is the advection time, c{sub 0} is the initial concentration in the medium, m{sub 0} is the 0th moment of the injection pulse; and t{sub a} 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, the authors tabulate g for many geometries, including several distributed (multirate) models of mass transfer. Using this expression they examine the behavior of late-time concentration for a number of mass transfer models. One key results is that if rate-limited mass transfer causes the BTC to behave as a power-law at late-time (i.e., c {approximately} t{sup {minus}k}), then the underlying density function of rate coefficients must also be a power-law with the form a{sup k{minus}}, as a {r_arrow}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 infinite, and also suggest an effective rate coefficient that is either undefined or changes as a function of observation time. They apply their analysis to breakthrough curves from Single-Well Injection-Withdrawal tests at the Waste Isolation Pilot Plant, New Mexico