Abstract
Non-Fickian diffusion has been increasingly documented in hydrology and modeled by promising time nonlocal transport models. While previous studies showed that most of the time nonlocal models are identical with correlated parameters, fundamental challenges remain in real-world applications regarding model selection and parameter definition. This study compared three popular time nonlocal transport models, including the multi-rate mass transfer (MRMT) model, the continuous time random walk (CTRW) framework, and the tempered time fractional advection–dispersion equation (tt-fADE), by focusing on their physical interpretation and feasibility in capturing non-Fickian transport. Mathematical comparison showed that these models have both related parameters defining the memory function and other basic-transport parameters (i.e., velocity v and dispersion coefficient D) with different hydrogeologic interpretations. Laboratory column transport experiments and field tracer tests were then conducted, providing data for model applicability evaluation. Laboratory and field experiments exhibited breakthrough curves with non-Fickian characteristics, which were better represented by the tt-fADE and CTRW models than the traditional advection–dispersion equation. The best-fit velocity and dispersion coefficient, however, differ significantly between the tt-fADE and CTRW. Fitting exercises further revealed that the observed late-time breakthrough curves were heavier than the MRMT solutions with no more than two mass-exchange rates and lighter than the MRMT solutions with power-law distributed mass-exchange rates. Therefore, the time nonlocal models, where some parameters are correlated and exchangeable and the others have different values, differ mainly in their quantification of pre-asymptotic transport dynamics. In all models tested above, the tt-fADE model is attractive, considering its small fitting error and the reasonable velocity close to the measured flow rate.
Highlights
Non-Fickian or anomalous transport, where the plume variance grows nonlinearly in time, has been documented extensively for solute transport in heterogeneous aquifers [1,2,3,4], soils [5,6,7], and rivers [8,9]
The use of the tempered time fractional advection–dispersion equation (tt-fADE) (10) and (11) to model mobile/immobile anomalous solute transport is motivated by four factors: (1) the equation governs the limits of known stochastic processes; (2) it describes a combination of first-order mass transfer models and reduces to known mobile/immobile equations in the integer order case; (3) the equation has tractable solutions that model the significant features of solute plume evolution in time and space; and (4) the equation is parsimonious, with no more parameters than the standard multi-rate mass transfer (MRMT) model (1) with multiple pairs of rate and capacity coefficients
There is no efficient way to directly measure the rate coefficient or capacity coefficient, which creates a challenge for relating MRMT model parameters to observable physical phenomena, and for providing information to MRMT models based on ancillary observations in heterogeneous media
Summary
Non-Fickian or anomalous transport, where the plume variance grows nonlinearly in time, has been documented extensively for solute transport in heterogeneous aquifers [1,2,3,4], soils [5,6,7], and rivers [8,9]. There are at least three popular time nonlocal transport models, which are the multi-rate mass transfer (MRMT) model [16,20], the hydrologic version of the continuous time random walk (CTRW) developed by Berkowitz and colleagues (see the extensive review in [21] and the mathematical version of CTRW in [22]), and the tempered time fractional advection–dispersion equation (tt-fADE) model [23] Some of these models have been compared theoretically. The applicability of those time nonlocal models is checked using laboratory experiments and field tests.
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