Application of Fractional Differential Equations for Modeling the Anomalous Diffusion of Contaminant from Fracture into Porous Rock Matrix with Bordering Alteration Zone

Fractional nutrient uptake model of plant roots

Most nutrient uptake problems are modeled by the convection-diffusion equation (CDE) abiding by Fick's law. Because nutrients needed by plants exist in the soil solution as a form of ions and the soil is a typical fractal structure of heterogeneity, it makes the solute transport appear anomalous diffusion in soil. Taking anomalous diffusion as a transport process, we propose time and space fractional nutrient uptake models based on the classic Nye-Tinker-Barber model. There does not appear apparent sub-diffusion of nitrate in the time fractional model until four months and the time fractional models are appropriate for describing long-term dynamics and slow sorption reaction; the space fractional model can capture super-diffusion in short term and it is suitable for describing nonlocal phenomena and daily variations driven by transpiration and metabolism; the anomalous diffusion more apparently appears near the root surface in the modeling simulation.

Generalized Skewed Model for Spatial-Fractional Advective–Dispersive Phenomena

The conventional mathematical model expressed by the advection–dispersion equation has been widely used to describe contaminant transport in porous media. However, studies have shown that it fails to simulate early arrival of contaminant, long tailing breakthrough curves and presents a physical scale-dependency of the dispersion coefficient. Recently, advances in fractional calculus allowed the introduction of fractional order derivatives to model several engineering and physical phenomena, including the anomalous dispersion of solute particles. This approach gives birth to the fractional advection–dispersion equation. This work presents new solutions to the fractional transport equation that satisfies the initial condition of constant solute injection in a semi-infinite medium. The new solution is derived based on a similarity approach. Moreover, laboratory column tests were performed in a Brazilian lateritic soil to validate the new solution with experimental data and compare its accuracy with the conventional model and other fractional solutions. The new solution outperforms the existing ones and reveals an interesting fractal-like scaling rule for the diffusivity coefficients.

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

Anomalous Solute Transport in a Cylindrical Two-Zone Medium with Fractal Structure

In this paper, a problem of anomalous solute transport in a coaxial cylindrical two-zone porous medium with fractal structure is posed and numerically solved. The porous medium is studied in the form of cylinder with two parts: macropore—with high permeability characteristics in the central part and micropore—with low permeability around it. Anomalous solute transport is modeled by differential equations with a fractional derivative. The solute concentration and pressure fields are determined. Based on numerical results, the influence of the fractional derivatives order on the solute transport process is analysed. It was shown that with a decrease in the order of the derivatives in the diffusion term of the transport equation in the macropore leads to a “fast diffusion” in both zones. Characteristics of the solute transport in both zones mainly depend on the concentration distribution and other hydrodynamic parameters in the macropore.

Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.

Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence.

A Two Dimensional Discrete Mollification Operator and the Numerical Solution of an Inverse Source Problem

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included.

Abiotic dechlorination in rock matrices impacted by long-term exposure to TCE

Field and laboratory tests were performed to evaluate the abiotic reaction of trichloroethene (TCE) in sedimentary rock matrices. Hydraulically conductive fractures, and the rock directly adjacent to the hydraulically conductive fractures, within a historically contaminated TCE bedrock aquifer were used as the basis for this study. These results were compared to previous work using rock that had not been exposed to TCE (Schaefer et al., 2013) to assess the impact of long-term TCE exposure on the abiotic dechlorination reaction, as the longevity of these reactions after long-term exposure to TCE was hitherto unknown. Results showed that potential abiotic TCE degradation products, including ethane, ethene, and acetylene, were present in the conductive fractures. Using minimally disturbed slices of rock core at and near the fracture faces, laboratory testing on the rocks confirmed that abiotic dechlorination reactions between the rock matrix and TCE were occurring. Abiotic daughter products measured in the laboratory under controlled conditions were consistent with those measured in the conductive fractures, except that propane also was observed as a daughter product. TCE degradation measured in the laboratory was well described by a first order rate constant through the 118-d study. Observed bulk first-order TCE degradation rate constants within the rock matrix were 1.3×10(-8) s(-1). These results clearly show that abiotic dechlorination of TCE is occurring within the rock matrix, despite decades of exposure to TCE. Furthermore, these observed rates of TCE dechlorination are expected to have a substantial impact on TCE migration and uptake/release from rock matrices.

Paradoxes of subdiffusive infiltration in disordered systems

Infiltration of diffusing particles from one material to another where the diffusion mechanism is either normal or anomalous is a widely observed phenomena. When the diffusion is anomalous we find interesting behavior: diffusion may lead to an averaged net drift x from one material to another even if all particles eventually flow in the opposite direction. Furthermore, x does not depend on the properties of the medium in which it is situated, indicating nonlocality of the process. Starting with an underlying continuous time random walk model we solve diffusion equations describing this problem. Similar drift against flow is found in the quenched trap model. We argue that such behavior is a general feature of diffusion in disordered systems.