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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6

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

Yong Zhang
  • State Key Lab. of Hydrology-Water Resources and Hydraulic Engineering College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
  • Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
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/ HongGuang Sun
  • State Key Lab. of Hydrology-Water Resources and Hydraulic Engineering College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
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/ Chunmiao Zheng
  • Guangdong Provincial Key Lab. of Soil and Groundwater Pollution Control, School of Environmental Science & Engineering Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
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Published Online: 2019-12-31 | DOI: https://doi.org/10.1515/fca-2019-0083

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

References

  • [1]

    E.E. Adams, L.W. Gelhar, Field study of dispersion in a heterogeneous aquifer 2. spatial moment analysis. Water Resour. Res. 28, No 12 (1992), 3293–3307.CrossrefGoogle Scholar

  • [2]

    B. Baeumer, Y. Zhang, R. Schumer, Incorporating super-diffusion due to sub-grid heterogeneity to capture non-Fickian transport. Ground Water 53, No 5 (2015), 699–708.CrossrefPubMedGoogle Scholar

  • [3]

    B. Baeumer, M. Kovács, M.M. Meerschaert, H. Sankaranarayanan, Boundary conditions for fractional diffusion. J. Comput. Appl. Math. 336 (2018), 408–424.CrossrefGoogle Scholar

  • [4]

    B. Baeumer, M. Kovács, H. Sankaranarayanan, Fractional partial differential equations with boundary conditions. J. Differ. Equ. 264, No 2 (2018), 1377–1410.CrossrefGoogle Scholar

  • [5]

    D.A. Benson, The fractional advection-dispersion equation: Development and application. Ph.D. Dissertation, Univ. of Nev., Reno (1998).Google Scholar

  • [6]

    D.A. Benson, M.M. Meerschaert, B. Baeumer, H.P. Scheffler, Aquifer operator scaling and the effect on solute mixing and dispersion. Water Resour. Res. 42 (2006), W01415; .CrossrefGoogle Scholar

  • [7]

    J.M. Boggs, S.C. Young, L.M. Beard, Field study of dispersion in a heterogeneous aquifer 1. Overview and site description. Water Resour. Res. 28, No 12 (1992), 3281–3291.CrossrefGoogle Scholar

  • [8]

    A. Chechkin, V.Y. Gonchar, J. Klafter, R. Metzler, L.V. Tanatarov, Lévy flights in a steep potential well. J. Stat. Phys. 115, No 516 (2004), 1505–1535.CrossrefGoogle Scholar

  • [9]

    W. Chen, L. Ye, H.G. Sun, Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, No. 5 (2010), 1614–1620.CrossrefGoogle Scholar

  • [10]

    W. Chen, G.F. Pang, A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J. Comput. Phys. 309, No 1 (2016); .CrossrefGoogle Scholar

  • [11]

    W. Chen, F. Wang, Singular boundary method using time-dependent fundamental solution for transient diffusion problems. Eng. Anal. Bound. Elem. 68, No 7 (2016), 115–123.CrossrefGoogle Scholar

  • [12]

    W. Chen, Y. Liang, X. Hei, Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1250–1261; ; https://www.degruyter.com/view/j/fca.2016.19.issue-5/issue-files/fca.2016.19.issue-5.xml.Crossref

  • [13]

    V.T. Chow, D.R. Maidment, L.W. Mays, Applied Hydrology. McGraw-Hill Publishing Companies (2013).Google Scholar

  • [14]

    O.A. Cirpka, A.J. Valocchi, Debates—Stochastic subsurface hydrology from theory to practice: Does stochastic subsurface hydrology help solving practical problems of contaminant hydrogeology? Water Resour. Res. 52 (2016); .CrossrefGoogle Scholar

  • [15]

    C.J.M. Cremer, I. Neuweiler, M. Bechtold, J. Vanderborght, Solute transport in heterogeneous soil with time-dependent boundary conditions. Vadose Zone J. 15, No 6 (2016); .CrossrefGoogle Scholar

  • [16]

    F. Delay, P. Ackerer, C. Danquigny, Simulating solute transport in porous or fractured formations using random walk particle tracking. Vadose Zone J. 4, No 2 (2005), 360–379.CrossrefGoogle Scholar

  • [17]

    M. Dentz, T. Le Borgne, A. Englert, B. Bijeljic, Mixing, spreading and reaction in heterogeneous media: A brief review. J. Contam. Hydrol. 120–121 (2011), 1–17.Google Scholar

  • [18]

    G.E. Fogg, Groundwater flow and sand body interconnectedness in a thick multiple-aquifer system. Water Resour. Res. 22 (1986), 679–694.CrossrefGoogle Scholar

  • [19]

    G.E. Fogg, Y. Zhang, Debates-stochastic subsurface hydrology from theory to practice: A geologic perspective. Water Resour. Res. 53, No 12 (2016), 9235–9245.Google Scholar

  • [20]

    R. Gorenflo, F. Mainardi, A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Soliton. Fract. 34, No 1 (2007), 87–103.CrossrefGoogle Scholar

  • [21]

    C.T. Green, Y. Zhang, B.C. Jurgens, J.J. Starn, M.K. Landon, Accuracy of travel time distribution (TTD) models as affected by TTD complexity, observation errors, and model and tracer selection. Water Resour. Res. 50, No 7 (2014), 6191–6213.CrossrefGoogle Scholar

  • [22]

    I. Gupta, A.M. Wilson, B.J. Rostron, Groundwater age, brine migration, and large-sale solute transport in the Alberta Basin. Canada. Geofluids 15 (2015), 608–620.CrossrefGoogle Scholar

  • [23]

    A.W. Harbaugh, MODFLOW-2005, The U.S. Geological Survey Modular Ground-Water Model - the Ground-Water Flow Process. U.S. Geological Survey Techniques and Methods 6-A16, Reston, Virginia (2005).Google Scholar

  • [24]

    K. Kang, S. Redner, Fluctuation-dominated kinetics in diffusion-controlled reactions. Phys. Rev. E 32, No 7 (1985), 435–447.CrossrefGoogle Scholar

  • [25]

    M. Karamouz, R. Kerachian, B. Zahraie, Monthly water resources and irrigation planning: case study of conjunctive use of surface and groundwater resources. J. Irrig. Drain Eng. 130, No 5 (2004), 391–402.CrossrefGoogle Scholar

  • [26]

    J.F. Kelly, H. Sankaranarayanan, M.M. Meerschaert, Boundary conditions for two-sided fractional diffusion. J. Comput. Phy. 376 (2019), 1089–1107.CrossrefGoogle Scholar

  • [27]

    J.C. Koch, R.C. Toohey, D.M. Reeves, Tracer-based evidence of heterogeneity in subsurface flow and storage within a boreal hillslope. Hydrol. Porcess. 31, No. 13 (2017), 2453–2463.CrossrefGoogle Scholar

  • [28]

    E.M. LaBolle, G.E. Fogg, A.F.B. Tompson, Random-walk simulation of transport in heterogeneous porous media: Local mass-conservation problem and implementation methods. Water Resour. Res. 32, No 3 (1996), 583–593.CrossrefGoogle Scholar

  • [29]

    E.M. LaBolle, J. Quastel, G.E. Fogg, J. Gravner, Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resour. Res. 36 (2000), 651–662.CrossrefGoogle Scholar

  • [30]

    E.M. LaBolle, G.E. Fogg, Role of molecular diffusion in contaminant migration and recovery in an alluvial aquifer system. Transp. Porous Media 42 (2001), 155–179.CrossrefGoogle Scholar

  • [31]

    E.M. LaBolle, RWHet: Random Walk Particle Model for Simulating Transport in Heterogeneous Permeable Media, Version 3.2, User’s Manual and Program Documentation. Univ. of Calif., Davis (2006).Google Scholar

  • [32]

    B.Q. Lu, Y. Zhang, H.G. Sun, C.M. Zheng, Lagrangian simulation of multi-step and rate-limited chemical reactions in multi-dimensional porous media. Water Sci. Eng. 11, No 2 (2018), 101–113.CrossrefGoogle Scholar

  • [33]

    B. Mandelbrot, J.W. van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, No 4 (1968), 422–437.CrossrefGoogle Scholar

  • [34]

    M. Marseguerra, A. Zoia, Monte Carlo investigation of anomalous transport in presence of a discontinuity and of an advection field. Physica A 377, No 2 (2007), 448–464.CrossrefGoogle Scholar

  • [35]

    M.M. Meerschaert, D.A. Benson, B. Baeumer, B., Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63 (2001), ID 021112; .CrossrefGoogle Scholar

  • [36]

    F. Obiri-Nyarko, S.J. Grajales-Mesa, G. Malina, An overview of permeable reactive barriers for in situ sustainable groundwater remediation. Chemosphere 111 (2014), 243–259.PubMedCrossrefGoogle Scholar

  • [37]

    L.J. Perez, J.J. Hidalgo, M. Dentz, Upscaling of mixing-limited bimolecular chemical reactions in Poiseuille flow. Water Resour. Res. 55, No 1 (2019), 249–269.CrossrefGoogle Scholar

  • [38]

    D.M. Reeves, D.A. Benson, M.M. Meerschaert, H.P. Scheffler, Transport of conservative solutes in simulated fracture networks: 2. Ensemble solute transport and the correspondence to operator-stable limit distribution. Water Resour. Res. 44 (2008), W05410; .CrossrefGoogle Scholar

  • [39]

    K.R. Rehfeldt, J.M. Boggs, L.W. Gelhar, Field study of dispersion in a heterogeneous aquifer 3. Geostatistical analysis of hydraulic conductivity. Water Resour. Res. 28, No 12 (1992), 3309–3324.CrossrefGoogle Scholar

  • [40]

    Salamon, P., D. Fernàndez-Garcia, J. J. Gómez-Hernández, Modeling mass transfer processes using random walk particle tracking. Water Resour. Res. 42 (2006), W11417; CrossrefGoogle Scholar

  • [41]

    X. Sanchez-Vila, D. Fernàndez-Garcia, A. Guadagnini, Interpretation of column experiments of transport of solutes undergoing an irreversible bimolecular reaction using a continuum approximation. Water Resour. Res. 46 (2010), W12510; .CrossrefGoogle Scholar

  • [42]

    W. Shao, Z. Yang, J. Ni, Y. Su, W. Nie, X. Ma, Comparison of single- and dual-permeability models in simulating the unsaturated hydro-mechanical behavior in a rainfall-triggered landslide. Landslides 15, No 12 (2018), 2449–2464.CrossrefGoogle Scholar

  • [43]

    E.R. Siirila, R.M. Maxwell, Evaluating effective reaction rates of kinetically driven solutes in large-scale statistically anisotropic media: human health risk implications. Water Resour. Res. 48 (2012), W04527.Google Scholar

  • [44]

    R. Sinha, M. Israil, D.C. Singhal, A hydrogeophysical model of the relationship between geoelectric and hydraulic parameters of anisotropic aquifers. Hydrogeol. J. 17 No 5 (2009), ID 495; .CrossrefGoogle Scholar

  • [45]

    A.Y. Sun, R.W. Ritzi, D.W. Sims, Characterization and modeling of spatial variability in a complex alluvial aquifer: implications on solute transport. Water Resour. Res. 44 (2008), W04402.Google Scholar

  • [46]

    H.G. Sun, A.L. Chang, Y. Zhang, W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications. Frac. Calc. Appl. Anal. 22, No 1 (2019), 27–59; ; https://www.degruyter.com/view/j/fca.2019.22.issue-1/issue-files/fca.2019.22.issue-1.xml.Crossref

  • [47]

    J.J.A. van Kooten, Groundwater contaminant transport including adsorption and first order decay. Stoch. Hydrol. Hydraul. 8, No 3 (1994), 185–205.CrossrefGoogle Scholar

  • [48]

    E. Vidal-Henriquez, V. Zykov, E. Bodenschatz, A. Gholami, Convective instability and boundary driven oscillations in a reaction-diffusion-advection model. Chaos 27 (2017), ID 103110; .CrossrefGoogle Scholar

  • [49]

    T. Vogel, H.H. Gerke, R. Zhang, M.Th. Van Genuchen, Modeling flow and transport in a two-dimensional dual-permeability system with spatially variable hydraulic properties. J. Hydro. 238, No 1-2 (2000), 78–89.CrossrefGoogle Scholar

  • [50]

    Y. Wang, S.A. Bradford, J. Simunek, Estimation and upscaling of dual-permeability model parameters for the transport of E. coli D21g in soils with preferential flow. J. Contam. Hydro. 159 (2014), 57–66.CrossrefGoogle Scholar

  • [51]

    Y. Zhang, D.A. Benson, M.M. Meerschaert, E.M. LaBolle, H.P. Scheffler, Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys. Rev. E 74 (2006), ID 026706; .CrossrefGoogle Scholar

  • [52]

    Y. Zhang, E.M. LaBolle, K. Pohlmann, Monte Carlo approximation of anomalous diffusion in macroscopic heterogeneous media. Water Resour. Res. 45 (2009), W10417; .CrossrefGoogle Scholar

  • [53]

    Y. Zhang, B. Baeumer, D.M. Reeves, A tempered multiscaling stable model to simulate transport in regional-scale fractured media. Geophy. Res. Lett. 37 (2010), L11405; .CrossrefGoogle Scholar

  • [54]

    Y. Zhang, C. Papelis, Particle-tracking simulation of fractional diffusion-reaction processes. Phys. Rev. E 84 (2011), 066704; .CrossrefGoogle Scholar

  • [55]

    Y. Zhang, C.T. Green, G.E. Fogg, Subdiffusive transport in alluvial settings: The influence of medium heterogeneity. Adv. Water Resour. 54 (2013), 78–99.CrossrefGoogle Scholar

  • [56]

    Y. Zhang, M.M. Meerschaert, B. Baeumer, E.M. LaBolle, Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resour. Res. 51 (2015), 6311–6337; .CrossrefGoogle Scholar

  • [57]

    Y. Zhang, C.T. Green, E.M. LaBolle, R.M. Neupauer, H.G. Sun, Bounded fractional diffusion in geological media: Definition and Lagrangian approximation. Water Resour. Res. 52, No 11 (2016), 8561–8577.CrossrefGoogle Scholar

  • [58]

    Y. Zhang, H.G. Sun, H.H. Stowell, M. Zayernouri, S.E. Hansen, A review of applications of fractional calculus in Earth system dynamics. Chaos Soliton. Frac. 102 (2017), 29–46.CrossrefGoogle Scholar

  • [59]

    Y. Zhang, B. Baeumer, L. Chen, D.M. Reeves, H.S. Sun, A fully subordinated linear flow model for hillslope subsurface stormflow. Water Resour. Res. 53 (2017), 3491–3504; .CrossrefGoogle Scholar

  • [60]

    Y. Zhang, M.M. Meerschaert, Particle tracking solutions of vector fractional differential equations: A review. In: Handbook of Fractional Calculus with Applications, Volume 3: Numerical Methods (2019), 275–285.Google Scholar

  • [61]

    C. Zheng, M. Bianchi, S.M. Gorelick, Lessons learned from 25 years of research at the MADE site. Ground Water 49, No 5 (2010), 649–662.PubMedGoogle Scholar

About the article

Received: 2019-07-12

Published Online: 2019-12-31

Published in Print: 2019-12-18


Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 6, Pages 1607–1640, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0083.

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