Mathematical equivalence between time-dependent single-rate and multirate mass transfer models

Abstract

The often observed tailing of tracer breakthrough curves is caused by a multitude of mass transfer processes taking place over multiple scales. Yet, in some cases, it is convenient to fit a transport model with a single‐rate mass transfer coefficient that lumps all the non‐Fickian observed behavior. Since mass transfer processes take place at all characteristic times, the single‐rate mass transfer coefficient derived from measurements in the laboratory or in the field vary with time $\omega (t)$. The literature review and tracer experiments compiled by Haggerty et al. (2004) from a number of sites worldwide suggest that the characteristic mass transfer time, which is proportional to $\omega (t)^{-1}$, scales as a power law of the advective and experiment duration. This paper studies the mathematical equivalence between the multirate mass transfer model (MRMT) and a time‐dependent single‐rate mass transfer model (t‐SRMT). In doing this, we provide new insights into the previously observed scale‐dependence of mass transfer coefficients. The memory function, $g(t)$, which is the most salient feature of the MRMT model, determines the influence of the past values of concentrations on its present state. We found that the t‐SRMT model can also be expressed by means of a memory function $\varphi (t,\tau )$. In this case, though the memory function is nonstationary, meaning that in general it cannot be written as $\varphi (t,-\tau )$. Nevertheless, the full behavior of the concentrations using a single time‐dependent rate $\omega (t)$ is approximately analogous to that of the MRMT model provided that the equality $\omega (t)=-dln{\frac {g(t)}{dt}}$ holds and the field capacity is properly chosen. This relationship suggests that when the memory function is a power law, $g(t)~t^{1-k}$, the equivalent mass transfer coefficient scales as $\omega (t)~t^{-1}$, nicely fitting without calibration the estimated mass transfer coefficients compiled by Haggerty et al. (2004).

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