Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

The forward Fractional Advection Dispersion Equation (FADE) provides a useful model for non‐Fickian transport in heterogeneous porous media. The space FADE captures the long leading tail, skewness, and fast spreading typically seen in concentration profiles from field data. This paper develops the corresponding backward FADE model, to identify source location and release time. The backward method is developed from the theory of inverse problems, and then explained from a stochastic point of view. The resultant backward FADE differs significantly from the traditional backward Advection Dispersion Equation (ADE) because the fractional derivative is not self‐adjoint and the probability density function for backward locations is highly skewed. Finally, the method is validated using tracer data from a well‐known field experiment, where the peak of the backward FADE curve predicts source release time, while the median or a range of percentiles can be used to determine the most likely source location for the observed plume. The backward ADE cannot reliably identify the source in this application, since the forward ADE does not provide an adequate fit to the concentration data.

A geologic perspective on stochastic subsurface hydrology offers insights on representativeness of prominent field experiments and their general relevance to other hydrogeologic settings. Although the gains in understanding afforded by some 30 years of research in stochastic hydrogeology have been important and even essential, adoption of the technologies and insights by practitioners has been limited, due in part to a lack of geologic context in both the field and theoretical studies. In general, unintentional, biased sampling of hydraulic conductivity (K) using mainly hydrologic, well‐based methods has resulted in the tacit assumption by many in the community that the subsurface is much less heterogeneous than in reality. Origins of the bias range from perspectives that are limited by scale and the separation of disciplines (geology, soils, aquifer hydrology, groundwater hydraulics, etc.). Consequences include a misfit between stochastic hydrogeology research results and the needs of, for example, practitioners who are dealing with local plume site cleanup that is often severely hampered by very low velocities in the very aquitard facies that are commonly overlooked or missing from low‐variance stochastic models or theories. We suggest that answers to many of the problems exposed by stochastic hydrogeology research can be found through greater geologic integration into the analyses, including the recognition of not only the nearly ubiquitously high variances of K but also the strong tendency for the good connectivity of the high‐K facies when spatially persistent geologic unconformities are absent. We further suggest that although such integration may appear to make the contaminant transport problem more complex, expensive and intractable, it may in fact lead to greater simplification and more reliable, less expensive site characterizations and models.

This study develops an explicit two‐step Lagrangian scheme based on the renewal‐reward process to capture transient anomalous diffusion with mixed retention and early arrivals in multidimensional media. The resulting 3‐D anomalous transport simulator provides a flexible platform for modeling transport. The first step explicitly models retention due to mass exchange between one mobile zone and any number of parallel immobile zones. The mobile component of the renewal process can be calculated as either an exponential random variable or a preassigned time step, and the subsequent random immobile time follows a Hyper‐exponential distribution for finite immobile zones or a tempered stable distribution for infinite immobile zones with an exponentially tempered power‐law memory function. The second step describes well‐documented early arrivals which can follow streamlines due to mechanical dispersion using the method of subordination to regional flow. Applicability and implementation of the Lagrangian solver are further checked against transport observed in various media. Results show that, although the time‐nonlocal model parameters are predictable for transport with retention in alluvial settings, the standard time‐nonlocal model cannot capture early arrivals. Retention and early arrivals observed in porous and fractured media can be efficiently modeled by our Lagrangian solver, allowing anomalous transport to be incorporated into 2‐D/3‐D models with irregular flow fields. Extensions of the particle‐tracking approach are also discussed for transport with parameters conditioned on local aquifer properties, as required by transient flow and nonstationary media.