Mathematical Equivalence Between Time-dependent Single-Rate and Multi-Rate Mass Transfer Models

The observed non-Fickian tailing in solute breakthrough curves is often caused by a multitude of mass transferprocesses taking place over multiple scales. Yet, in some cases it is convenient to fit a transport model with asingle-rate mass transfer coefficient that lumps all the non-Fickian observed behavior. Since mass transfer processestake place at all characteristic times, the single-rate mass transfer coefficient derived from measurements in thelaboratory or in the field vary with time, β(t). In this context, we present a mathematical equivalence between the Multi-RateMass Transfer Model (MRMT) and an effective time-dependent single-rate mass transfer model (t-SRMT). In doing this, we provide newinsights 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 onits present state. We found that the t-SRMT model can also be expressed by means of a memory functionφ(t,s). In this case though the memory function is non-stationary, meaning that in general it cannot bewritten as φ(t-s). Nevertheless, the concentration breakthrough curves obtained using an effective single time-dependentrate β(t) is analogous to that of the MRMT model provided that a simple equality holds. Thisrelationship suggests that when the memory function is a power law, g(t)~ t1-m, the equivalent mass transfercoefficient scales as β(t)~ t-1. A result that explains the scaling exponent of the mass transfer coefficient reported by the literature review and tracer experiments of Haggerty etal. [2004] of -0.94.