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Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application

  • Yong Zhang EMAIL logo , HongGuang Sun and Chunmiao Zheng
Published/Copyright: December 31, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 6

Abstract

Fractional-derivative models (FDMs) are promising tools for characterizing non-Fickian transport in natural geological media. Hydrologic applications of FDMs, however, have been limited in the last two decades, due to the lack of feasible models and solvers to quantify multi-dimensional anomalous diffusion for pollutants in bounded aquifers. This study develops and applies FDM tools to capture vector fractional dispersion for both conservative and reactive pollutants in fractional Brownian motion (fBm) random fields with bounded domains. A d-dimensional anisotropic fBm field for hydraulic conductivity (K) is first generated numerically. A particle-tracking based, fully Lagrangian solver is then developed to approximate particle dynamics in the fBm K fields under various boundary conditions, where the governing equation is the vector FDM subordination to regional flow. Numerical experiments show that the Lagrangian solver can combine nonlocal anomalous transport and local aquifer properties to quantify pollutant transport in bounded aquifers. Application analyses further reveal that the K correlation can significantly enhance the spreading of conservative pollutant particles, and increase the reaction rate by enhancing the mobility and mixing of reactant particles undergoing bimolecular reactions.

Extension of the Lagrangian solver is also discussed, including modeling transient flow, generalizing boundary conditions, and capturing complex chemical reactions. This study therefore provides the hydrologic community an efficient Lagrangian solver to model reactive anomalous transport in bounded anisotropic aquifers with any dimension, size, and boundary conditions.

MSC 2010: Primary 26A33; Secondary 34A08; 34A34; 34K28; 35R11; 60G22; 65L10; 82C70; 86A05
Key Words and Phrases: vector fractional-derivative model; Lagrangian solver; fractional Brownian motion; bimolecular reaction; bounded domain

This paper is dedicated to the memory of the late Professor Wen Chen

Appendix A. Lagrangian solution of the multi-scaling fADE (2.8)

The M-fADE (2.8) can define three types of vector diffusion, to capture complex multi-dimensional transport in geological media. The three types of vector diffusion exhibit different relationships between the mixing measure M and the scaling matrix H∗.

Type-1 vector diffusion: Eigenvectors in H∗ differ from the jumping angles in M(dθ). In some heterogeneous porous media, groundwater flow and contaminant transport do not need to follow the same directions. For example, water can flow along depositional stratigraphic (strike or dip) directions which can be defined by the scaling directions (i.e., eigenvectors) in H∗, while the contaminants move along preferential pathways formed by interconnected high-permeability deposits which can be defined by the angles dθ in the mixing measure M. This leads to model (2.8) with (usually orthogonal) eigenvectors in H∗ and an arbitrary mixing measure M(dθ); see [51] for examples.

Type-2 vector diffusion: Eigenvectors in H∗ are consistent with the jumping angles in M(dθ). For example, in discrete fracture networks (DFNs), water and contaminant move along the DFN’s orientations [38], which can be captured by model (2.8) with H∗ containing nonorthogonal eigenvectors and M(dθ) concentrated on eigenvectors. A specific example of type-2 vector diffusion is the classical 2-d fADE:

LtC=−∂∂xvxC−∂∂yvyC+∂∂xDx∂αx−1C∂xαx−1+∂∂yDy∂αy−1C∂yαy−1,(6.1)

where Lt denotes the temporal operator, and the space indexes 1 < αx < 2 and 1 < αy < 2. Here the Riemann-Liouville fractional derivative is used for the space fractional derivative, to have correct solutions in bounded domains (while the Caputo fractional derivative leads to negative solutions at the boundary) [3]. Note that Professor W. Chen had also developed the Eulerian approaches to solve the vector fADE (6.1). For example, Chen et al. [9] developed the meshless Kansa method to solve anomalous diffusion in irregular flow fields, Chen and Wang [11] applied the singular boundary method to solve transient diffusion, and Chen et al. [12] also solved anomalous diffusion described by a structural derivative model. Hence, this study extends Professor Chen’s work using the Lagrangian solver.

Type-3 vector diffusion: Super-diffusion with a streamline-dependent mixing measure M. The directions of pollutant dispersion may not be fixed, but rather change in space due to the local variation of preferential pathways (note this assumption is widely used in hydrological models [21]), which can be captured by model (2.8) with a streamline-dependent M.

The Lagrangian solver for the M-fADE (2.8) is similar to that for the S-fADE (2.1), except for step 2 (random spatial jumps). Here the random displacement can be calculated using the following multi-scaling compound Poisson process for the i-th jump [51]:

Z→(τ)=∑[τ/dM]RiH∗⋅θ→i,(6.2)

where Z⃗(τ) is the random walker’s location at the operational time τ, [τ/dM] is the number of jumps by time τ with the step size dM, and Ri is the random length of the i-th jump along the independent jumping direction θi:

R1/αk=D(xk)cos⁡παk2dM1/αkdLαkλk+Θ(αk−1)cos⁡π(αk−1)2dM1/αk−1dLαk−1λk,(6.3)

where R1/αk denotes the random displacement along the k-th eigenvector of H∗; Θ = ∂D/∂xk, –∂D/∂xk, or 0 if ∂D/∂xk > 0, < 0, and 0, respectively; and dLαkλk and dLαk−1λk denote the αk- and the αk – 1-order, tempered stable random variables with the maximum skewness, scale one, zero shift, and truncation parameter λk, respectively.

Step 2 expressed by Eq. (6.2) and Eq. (6.3) can be modified for the three types of anomalous transport discussed above to achieve the best computational efficiency in the Lagrangian solver. For type-1 transport, components of the jump vector described by (6.3) are dependent; while for type-2 transport, the jump along each eigenvector is independent to the others, resulting in a simplified Lagrangian solver. For type-3 transport, the random displacement (6.3) can be projected to the adjacent streamline.

Acknowledgements

The work was partially supported by the National Natural Science Foundation of China (under grants 41931292, 41330632, and 11972148) and Guangdong Provincial Key Laboratory of Soil and Groundwater Pollution Control. This paper does not necessary reflect the view of the funding agencies.

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Received: 2019-07-12
Published Online: 2019-12-31
Published in Print: 2019-12-18

© 2019 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
  4. Survey Paper
  5. State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
  6. An investigation on continuous time random walk model for bedload transport
  7. Tutorial Survey
  8. Porous functions
  9. Research Paper
  10. A time-space Hausdorff derivative model for anomalous transport in porous media
  11. High-order algorithms for riesz derivative and their applications (IV)
  12. Mass-conserving tempered fractional diffusion in a bounded interval
  13. Dispersion analysis for wave equations with fractional Laplacian loss operators
  14. Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
  15. Some further results of the laplace transform for variable–order fractional difference equations
  16. Robust stability analysis of LTI systems with fractional degree generalized frequency variables
  17. Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
https://doi.org/10.1515/fca-2019-0083
Keywords for this article
vector fractional-derivative model; Lagrangian solver; fractional Brownian motion; bimolecular reaction; bounded domain

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA special issue – In memory of late professor Wen Chen (FCAA–Volume 22–6–2019)
  4. Survey Paper
  5. State-of-art survey of fractional order modeling and estimation methods for lithium-ion batteries
  6. An investigation on continuous time random walk model for bedload transport
  7. Tutorial Survey
  8. Porous functions
  9. Research Paper
  10. A time-space Hausdorff derivative model for anomalous transport in porous media
  11. High-order algorithms for riesz derivative and their applications (IV)
  12. Mass-conserving tempered fractional diffusion in a bounded interval
  13. Dispersion analysis for wave equations with fractional Laplacian loss operators
  14. Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application
  15. Some further results of the laplace transform for variable–order fractional difference equations
  16. Robust stability analysis of LTI systems with fractional degree generalized frequency variables
  17. Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network
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