A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

Abstract

Fractional advection–dispersion equations (FADEs) have been widely used in hydrological research to simulate the anomalous solute transport in surface and subsurface water. However, a large gap still exists between real‐world application (i.e., being a prediction tool) and theoretical FADEs. To better understand this disparity, the FADEs are firstly reviewed from the perspective of fractional‐in‐time and fractional‐in‐space, as well as the anomalous characteristics described by those functions. Then, challenges for the application of FADEs are summarized, including the theoretical gap of FADEs that needs multidisciplinary efforts to fill, extensive requirements for computation techniques and mathematical knowledge to apply the FADEs, the poor predictability for most parameters in the FADEs, and the limitations for collecting geologic information of flow fields. Then, some suggestions are given for future work, such as developing excellent code sets and mature simulation software with a friendly interface. This kind of work would alleviate the computation workload of hydrologists, especially those without coding expertise. Summarizing and determining the value ranges of the important parameters are needed (e.g., the order of the fractional derivative), through the extensive field, laboratory, and numerical experiments rather than blindly using their mathematical ranges. It is also needed to perform global sensitivity analysis for the FADEs. Meanwhile, comprehensive comparison work is necessary for suggesting a model suitable for a specific problem. Last but not least, further research is clearly needed to establish a link between nonlocal parameters and the heterogeneity property to develop more efficient fractional order partial differential equations. This article is categorized under: Science of Water > Methods Science of Water > Water Quality

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

“…A simple and fairly general FADE for subsurface transport under the influence of both advection and anomalous diffusion is the one-dimensional advection and spacetime-fractional diffusion equation with constant coefficients (see, e.g., [223]):…”

“…Such media feature heterogeneities that are either difficult or impossible to observe, and hence cannot be described with certainty at all relevant scales and locations. Moreover, even when the environment's microstructure can be captured, numerical simulations of appropriate PDE models such as systems of advection-diffusion equations may be infeasibly expensive if conducted at fully-resolved small scales [223]. In fact, the same types of equations that are accurate at small scales do not extrapolate and predict solutes' behavior at larger scales, due to the appearance of "anomalous", or "non-Fickian" behavior [21,22,125,160].…”

A co-spectral budget model links turbulent eddies to suspended sediment concentration in channel flows

“…Another revision to equation 10 is to re-cast turbulent transport in fractional derivatives to emphasize its non-Fickian aspect (Nie et al, 2017). In this approach, the fractional order becomes a parameter that must be determined from experiments depending on how SS trajectories deviate from Brownian trajectories (Sun et al, 2020). In practice, the order of the fractional derivative is set as a 'free' parameter and must implicitly include the Sc effect.…”

Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves

“…Making accurate large-scale predictions of solute transport in the subsurface is critically important for the efficient management of water resources [25,26] as well as petroleum production, particularly in enhanced oil-recovery (EOR) applications [1,14,24,21]. Subsurface transport is a highly complex phenomenon as it takes place in environments that contain heterogeneities at all scales, requiring the use of fine-scale models at the smallest scales.…”