河道溶质输运过程的截断型分数阶导数建模研究
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摘要
河道中溶质在低速区和高速区内的复杂交互作用是导致溶质发生反常输运现象的主要原因之一。截断型分数阶导数模型将复杂流场中溶质输运过程视为低速滞留区和高速流动区的耦合过程,并引入截断系数描述溶质浓度由幂率衰减到指数衰减的过程。考察了不同条件下模型参数对截断型分数阶导数模型模拟结果的影响。并利用此模型模拟相同实验条件下不同粒子的不同拖尾行为,对比实验观测数据发现,截断型分数阶导数模型能够较好的模拟衰减过程变化的拖尾现象。
The intricate interaction of solute in streams within the active channel and the retention zone is one of the main reasons leading to anomalous solute transport. Tempered fractional advection-dispersion model was introduced to characterize the solute transport process in complex flow field as the coupling process of the low-speed retention zone and the high-speed flow zone. In this model,a truncation parameter was employed to describe the transition from anomalous transport( power law decay) to normal transport( exponential decay) with time. In this paper,the influence of the parameters on the numerical results of the tempered fractional derivative advection-dispersion model under different conditions was investigated. Moreover,tailing behaviors of different solutes under the same experimental conditions were simulated by the tempered fractional model.It was found that the tempered fractional model effectively characterized the tailing behaviors in attenuation process.
The intricate interaction of solute in streams within the active channel and the retention zone is one of the main reasons leading to anomalous solute transport. Tempered fractional advection-dispersion model was introduced to characterize the solute transport process in complex flow field as the coupling process of the low-speed retention zone and the high-speed flow zone. In this model,a truncation parameter was employed to describe the transition from anomalous transport( power law decay) to normal transport( exponential decay) with time. In this paper,the influence of the parameters on the numerical results of the tempered fractional derivative advection-dispersion model under different conditions was investigated. Moreover,tailing behaviors of different solutes under the same experimental conditions were simulated by the tempered fractional model.It was found that the tempered fractional model effectively characterized the tailing behaviors in attenuation process.
引文
[1]孙洪广,陈文.复杂地下含水层中溶质扩散过程的变导数分数阶扩散模型[C]//中国力学大会.西安,2013.
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[13]Cai W,Chen W,Xu W X.Fractional modeling of Pasternak-type viscoelastic foundation[J].Mechanics of Time-Dependent Materials,2017,21(1):119-131.
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[21]孙洪广,陈文,蔡行.空间分数阶导数“反常”扩散方程数值算法的比较[J].计算物理,2009,26(5):719-724.
[22]Sun H G,Zhang Y,Chen W,et al.Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media[J].Journal of Contaminant Hydrology,2014,157(3):47-58.
[23]Zhang Y.Comparison of the fractional advection-dispersion equation and the fractional Fokker-Planck equation:Fractional dynamics and real-world applicability[J].Journal of Nanjing University,2011,47(3):265-275.
[24]Metzler R,Klafter J.The random walk's guide to anomalous diffusion:A fractional dynamics approach[J].Physics Reports,2000,339(1):1-77.
[25]Liu F W,Anh V V,Turner I,et al.Time fractional advectiondispersion equation[J].Journal of Applied Mathematics and Computing,2003,13(1):233-245.
[26]Zhang Y,Chen L,Reeves D M.A fractional-order temperedstable continuity model to capture surface water runoff[J].Journal of Vibration&Control,2016,22(8):1993-2003.
[27]Sabzikar F,Meerschaert M M,Chen J.Tempered fractional calculus[J].Journal of Computational Physics,2015,293(C):14-28.
[28]Zhang Y,Green C T,Baeumer B.Linking aquifer spatial properties and non-Fickian transport in mobile-immobile like alluvial settings[J].Journal of Hydrology,2014,512(9):315-331.
[29]Xia Y,Wu J C,Zhang Y.Tempered time-fractional advectiondispersion equation for modeling non-Fickian transport[J].Advances in Water Science,2013,24(3):349-357.
[30]Sun H G,Chen W,Li C,et al.Finite difference schemes for variable-order time fractional diffusion equation[J].International Journal of Bifurcation and Chaos,2012,22(4):1250085.
[31]Schumer R,Benson D A,Meerschaert M M,et al.Fractal mobile/immobile solute transport[J].Water Resources Research,2003,39(10):1296.
[32]Yu C,Rafael M C,Gao B,et al.Effects of ionic strength,particle size,flow rate,and vegetation type on colloid transport through a dense vegetation saturated soil system:Experiments and modeling[J].Journal of Hydrology,2013,499(9):316-323.
[2]Ganti V,Meerschaert M M,Foufoula-Georgiou E,et al.Normal and anomalous diffusion of gravel tracer particles in rivers[J].Journal of Geophysical Research Earth Surface,2010,115(F2):923-928.
[3]Young P,Wallis S.Solute transport and dispersion in channels[J].Channel Network Hydrology,1993,129-173.
[4]Kim S,Kavvas M L.Generalized Fick’s law and fractional ADE for pollution transport in a river:Detailed derivation[J].Journal of Hydrologic Engineering,2006,11(1):80-83.
[5]Cussler E L.Diffusion:Mass Transfer in Fluid Systems[M].Cambridge:Cambridge University Press,2009.
[6]孙洪广,常爱莲,陈文,等.反常扩散:分数阶导数建模及其在环境流动中的应用[J].中国科学:物理学力学天文学,2015,45(10):104702.
[7]Weickert J.A review of nonlinear diffusion filtering[J].Lecture Notes in Computer Science,1997,1252:1-28.
[8]Burgers J M.The nonlinear diffusion equation[J].Nonlinear Diffusion Equations,2006,152(4):608-615.
[9]Mangat P S,Molloy B T.Prediction of long term chloride concentration in concrete[J].Materials&Structures,1994,27(6):338-346.
[10]Chen W,Zhang J J,Zhang J Y.A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures[J].Fractional Calculus&Applied Analysis,2013,16(1):76-92.
[11]Benson D A,Tadjeran C,Meerschaer T M M,et al.Radial fractional-order dispersion through fractured rock[J].Water Resources Research,2004,40(12):87-87.
[12]Benson D A,Wheatcraft S W,Meerschaert M M.The fractionalorder governing equation of Lévy Motion[J].Water Resources Research,2000,36(6):1413-1424.
[13]Cai W,Chen W,Xu W X.Fractional modeling of Pasternak-type viscoelastic foundation[J].Mechanics of Time-Dependent Materials,2017,21(1):119-131.
[14]Chen W,Sun H G,Zhang X D,et al.Anomalous diffusion modeling by fractal and fractional derivatives[J].Computers&Mathematics with Applications,2010,59(5):1754-1758.
[15]Meerschaert M M,Sabzikar F,Phanikumar M S,et al.Tempered fractional time series model for turbulence in geophysical flows[J].Journal of Statistical Mechanics Theory&Experiment,2014(9):09023.
[16]Meerschaert M M,Zhang Y,Boris B.Tempered anomalous diffusion in heterogeneous systems[J].Geophysical Research Letters,2008,35(17):L17403.
[17]Podlubny I.Fractional Differential Equations[C]∥Mathematics in Science and Engineering,1999.
[18]Wei S,Chen W,Hon Y C.Characterizing time dependent anomalous diffusion process:A survey on fractional derivative and nonlinear models[J].Physica A Statistical Mechanics&Its Applications,2016,462:1244-1251.
[19]Zhang Y,Baeumer B,Reeves D M.A tempered multiscaling stable model to simulate transport in regional-scale fractured media[J].Geophysical Research Letters,2010,37(37):11405.
[20]Zhang Y,Benson D A,Reeves D M.Time and space nonlocalities underlying fractional-derivative models:Distinction and literature review of field applications[J].Advances in Water Resources,2009,32(4):561-581.
[21]孙洪广,陈文,蔡行.空间分数阶导数“反常”扩散方程数值算法的比较[J].计算物理,2009,26(5):719-724.
[22]Sun H G,Zhang Y,Chen W,et al.Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media[J].Journal of Contaminant Hydrology,2014,157(3):47-58.
[23]Zhang Y.Comparison of the fractional advection-dispersion equation and the fractional Fokker-Planck equation:Fractional dynamics and real-world applicability[J].Journal of Nanjing University,2011,47(3):265-275.
[24]Metzler R,Klafter J.The random walk's guide to anomalous diffusion:A fractional dynamics approach[J].Physics Reports,2000,339(1):1-77.
[25]Liu F W,Anh V V,Turner I,et al.Time fractional advectiondispersion equation[J].Journal of Applied Mathematics and Computing,2003,13(1):233-245.
[26]Zhang Y,Chen L,Reeves D M.A fractional-order temperedstable continuity model to capture surface water runoff[J].Journal of Vibration&Control,2016,22(8):1993-2003.
[27]Sabzikar F,Meerschaert M M,Chen J.Tempered fractional calculus[J].Journal of Computational Physics,2015,293(C):14-28.
[28]Zhang Y,Green C T,Baeumer B.Linking aquifer spatial properties and non-Fickian transport in mobile-immobile like alluvial settings[J].Journal of Hydrology,2014,512(9):315-331.
[29]Xia Y,Wu J C,Zhang Y.Tempered time-fractional advectiondispersion equation for modeling non-Fickian transport[J].Advances in Water Science,2013,24(3):349-357.
[30]Sun H G,Chen W,Li C,et al.Finite difference schemes for variable-order time fractional diffusion equation[J].International Journal of Bifurcation and Chaos,2012,22(4):1250085.
[31]Schumer R,Benson D A,Meerschaert M M,et al.Fractal mobile/immobile solute transport[J].Water Resources Research,2003,39(10):1296.
[32]Yu C,Rafael M C,Gao B,et al.Effects of ionic strength,particle size,flow rate,and vegetation type on colloid transport through a dense vegetation saturated soil system:Experiments and modeling[J].Journal of Hydrology,2013,499(9):316-323.