Fractional and fractal derivative models for transient anomalous diffusion: Model comparison

Abstract

Transient anomalous diffusion characterized by transition between diffusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian diffusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional diffusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), breakthrough curves (exhibiting nuance in diffusive state transition), and possible hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures transient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to capture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications.

Introduction

Fractional calculus and fractional-derivative models (FDE) are promising tools for describing non-Fickian or anomalous transport observed in various heterogeneous systems, including porous media, fractured networks, turbulent flow, and complex materials [1], [2], [3]. Anomalous diffusion due to intrinsic multi-scale geometrical/physical heterogeneity of natural geological formations cannot be efficiently quantified by Fick’s law and the standard advection-dispersion equation (ADE) [4], [5], [6]. Both the FDE and fractal derivative ADE models (which can describe power-law and stretched-exponential decay, respectively) have been successfully applied to interpret heavy-tailed dynamics in many fields [7], [8], [9]. After decades of FDE application, two trends have emerged for the future development and application of mathematical models in natural sciences, especially the hydrologic sciences where non-Fickian transport has been well documented [4], [5]. First, complex real-world non-Fickian transport has been identified with field measurements, with greater nuance for detecting diffusion. Second, many FDEs (and also fractal models) have been proposed, but systematic model comparison remains scarce.

Transient anomalous diffusion is the representative example of the above two trends. In real-world complex systems, mechanisms that can fluctuate spatiotemporally tend to cause transitions between diffusive states (i.e., super-diffusion, sub-diffusion, and normal-diffusion) for mass, momentum, or energy movement, a phenomenon called “transient diffusion” [10], [11]. For instance, non-Fickian transport in natural geological media (rivers, soils, land surface, and aquifers) depends on the medium’s physical heterogeneity, soil mineralogy, flow properties (such as flow rate and direction, and boundary/initial conditions), and tracer chemical properties (concentration, sorption, activity, pH, temperature). These driving forces do not necessarily remain stable, but rather change in space and/or time under natural conditions. For example, groundwater flow in natural aquifers is not constant but naturally transient, which can result in the time-dependent scaling of solute transport or transient diffusion [5]. It is therefore necessary to develop efficient FDEs to capture transient diffusion, as required by many practical applications such as the pump-and-treat remediation of groundwater resources. However, multiple mathematical models have been proposed to capture transient diffusion, although the methodologies have not been systematically evaluated, motivating this study.

Four types of nonlocal/local transport models have been proposed to quantify transient diffusion. First, Chechkin et al. [12] proposed a distributed-order fractional diffusion equation (D-FDE) model to describe retarding sub-diffusion and accelerating super-diffusion. Second, Meerschaert et al. [10] introduced a tempered fractional diffusion equation (T-FDE) model to capture the slow convergence of sub-diffusion to its Gaussian asymptote for passive tracers in heterogeneous media, and further validated it against numerical simulations and field measurements. Third, Sun et al. [13] offered a unified discussion on the variable-order fractional diffusion equation (V-FDE) model and explored the physical origin causing the index to change with time, space, concentration or other independent quantities. Fourth, Liu et al. [14] introduced a variable-order fractal (Hausdorff) derivative diffusion equation (V-HDE) model to describe time-dependent diffusion and further compared it with the V-FDE model in characterizing anomalous diffusion. These four models have attracted sustained attention in transient diffusion modeling and engineering applications [15], [16], [17], [18], [19]. However, different models and theories might be formulated for specific transition sequences, and therefore might be valid for different types of transient diffusion.

This study aims to evaluate the above models by comparing their mean squared displacement (MSD) (which is the criteria for diffusion states in transient diffusion), breakthrough curves (BTCs) (which can exhibit changes in transient diffusion characteristics), and possible hydrogeologic origin (linkage to medium property), with the expectation to offer a guideline for real-world applications. The rest of this paper is organized as follows. The basic idea, governing equation, and major property of the above four models are presented in Section 2. A detailed comparison and further discussion characterizing transient diffusion are presented in Section 3. A conclusion is drawn in Section 4.