分形结构中介质渗透性对扩散过程的影响
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摘要
土壤孔隙由于其结构的不均匀特征使得溶质的运移规律不符合经典的菲克定律。为了研究溶质在土壤等不均匀结构中的扩散行为,建立了Sierpinski分形地毯模型模拟非均匀性的土壤结构,并通过投放大量粒子进行随机行走模拟,目的是研究土壤中污染物等溶质的扩散规律。选用同一种Sierpinski分形地毯模型,并取相同的起始投放点位置,粒子数量以及运动步数,以不同规则下粒子的随机运动模拟为手段,探索不同介质条件下溶质的运动规律。根据分形结构中溶质位移与时间的二阶矩关系,通过数值模拟、对比两种条件下的Hurst值及其影响因素,从而得到分形结构中粒子的扩散规律。
The structural of soil has been proved to be a heterogeneous one,which causes the solute transport behavior is quite different from the ideal homogeneous medium and violates Fick's law. To explore the transport behavior in fractured medium,a Sierpinski fractal carpet model was established to represent the heterogeneity of soil structure. Random particle motion simulation under different rules were applied to characterize the random motion of solute particles under different medium conditions based on Sierpinski fractal carpet model with same starting point,particle number and number of moving steps. Finally,the diffusion behavior in the fractal structure is explored according to the second-order moment expression of displacement in the fractal structure. The influencing factor Hurst are obtained by comparison in the two cases.
The structural of soil has been proved to be a heterogeneous one,which causes the solute transport behavior is quite different from the ideal homogeneous medium and violates Fick's law. To explore the transport behavior in fractured medium,a Sierpinski fractal carpet model was established to represent the heterogeneity of soil structure. Random particle motion simulation under different rules were applied to characterize the random motion of solute particles under different medium conditions based on Sierpinski fractal carpet model with same starting point,particle number and number of moving steps. Finally,the diffusion behavior in the fractal structure is explored according to the second-order moment expression of displacement in the fractal structure. The influencing factor Hurst are obtained by comparison in the two cases.
引文
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14 Voller V R.A direct simulation demonstrating the role of special heterogeneity in determining anomalous diffusive transport[J].Water Resources Research,2015,51(4):2119-2127
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16张东辉.多孔介质扩散、导热、渗流分形模型的研究[D].南京:东南大学,2003Zhang Donghui.Diffusion percolation and heat conduction in fractal porpus media[D].Nanjing:Southeast China University,2003
17 Sahimi M,Hughes B D,Scriven L E,et al.Dispersion in flow through porous media-I.One-phase flow[J].Chemical Engineering Science,1986,41(8):2123-2136
18 Coppens M O,Froment G F.Diffusion and reaction in a fractal catalyst pore-I.Geometrical aspects[J].Chemical Engineering Science,1995,50(6):1013-1026
19 Tarafdar S,Franz A,Schulzky C,et al.Modelling porous structures by repeated Sierpinski carpets[J].Physica a Statistical Mechanics&Its Applications,2001,292(1):1-8
20李小川,施明恒.多孔介质通道的分形随机行走描述[C]//传热传质学学术会议.北京:中国工程热物理学会,2009:477-479Li Xiaochuan,Shi Mingheng.Fractal random walk description of porous media channels[C]//Academic Conference on Heat and Mass Transfer.Beijing:Chinese Society of Engineering Thermophysics,2009:477-479
21贝尔.多孔介质流体动力学[M].李竞生,陈崇希,译.北京:中国建筑工业出版社,1983Bear J.Dynamics of fluids in porous media[M].Li Jingsheng,Chen Chongxi,translated.Beijing:China Building Industry Press,1983
22 Chechkin A V,Gorenflo R,Sokolov I M.Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations[J].Physical Review E Statistical Nonlinear&Soft Matter Physics,2002,66:046129
23 Podlubny I.Fractional differential equation[M].San diego,Academic Press,1999
24 Gouyet J F.Physics and fractal structures[M].Berlin:SpringerVerlag,1996
2 Kendall D.Principles of random walk[J].Journal of the London Mathematical Society,1967,1(1):377-378
3 Manocha H L,Sharma B L.Some formulae by means of fractional derivatives[J].Compositio Mathematica,1967,18:229-234
4 Montroll E W,Weiss G H.Random walks on lattices.II[J].Journal of Mathematical Physics,1965,6(2):167-181
5 Mandelbrot B B.The fractal geometry of nature[M].San Francisco:WH Freeman,1982:30-35
6 Katz A J,Thompson A H.Fractal sandstone pores:Implications for conductivity and pore formation[J].Physical Review Letters,1985,54(12):1325-1328
7 Anderson A N.Soil mass,surface,and spectral fractal dimensions estimated from thin section photographs[J].Soil Science Society of America Journal,1996,60(4):962-969
8 Kaye B H.分形漫步[M].徐新阳,译.沈阳:东北工学院出版社,1994Kaye B H.Fractal walk[M].Xu Xinyang,trans lated.Shenyang:Northeast Institute of Technology Press,1994
9 Abbasi M H,Evans J W,Abramson I S.Diffusion of gases in porous solids:Monte Carlo simulations in the Knudsen and ordinary diffusion regimes[J].Aiche Journal,2010,29(4):617-624
10刘伟,范爱武,黄晓明.多孔介质传热传质理论与应用[M].北京:科学出版社,2006Liu Wei,Fan Aiwu,Huang Xiaoming.Theory and application of heat and mass transfer in porous media[M].Beijing:Science Press,2006
11 Mukhopadhyay S,Cushman J H.Diffusive transport of volatile pollutants in nonaqueous-phase liquid contaminated soil:A fractal model[J].Transport in Porous Media,1998,30(2):125-154
12陈文,孙洪广,李西成.力学与工程问题的分数阶导数建模[M].北京:科学出版社,2010Chen Wen,Sun Hongguang,Li Xicheng.Fractional derivative modeling for mechanics and engineering problems[M].Beijing:Science Press,2010
13包景东.分数布朗运动和反常扩散[J].物理学进展,2005,25(4):359-367Bao Jingdong.Fractional brown motion and anomalous diffusion[J].Progress in Physics,2005,25(4):359-367
14 Voller V R.A direct simulation demonstrating the role of special heterogeneity in determining anomalous diffusive transport[J].Water Resources Research,2015,51(4):2119-2127
15张东辉,施明恒,金峰,等.分形多孔介质的粒子扩散特点[J].工程热物理学报,2004,25(5):825-827Zhang Donghui,Shi Mingheng,Jin Feng,et al.Diffusion characteristic in fractal porous media[J].Journal of Engineering Thermophysics,2004,25(5):825-827
16张东辉.多孔介质扩散、导热、渗流分形模型的研究[D].南京:东南大学,2003Zhang Donghui.Diffusion percolation and heat conduction in fractal porpus media[D].Nanjing:Southeast China University,2003
17 Sahimi M,Hughes B D,Scriven L E,et al.Dispersion in flow through porous media-I.One-phase flow[J].Chemical Engineering Science,1986,41(8):2123-2136
18 Coppens M O,Froment G F.Diffusion and reaction in a fractal catalyst pore-I.Geometrical aspects[J].Chemical Engineering Science,1995,50(6):1013-1026
19 Tarafdar S,Franz A,Schulzky C,et al.Modelling porous structures by repeated Sierpinski carpets[J].Physica a Statistical Mechanics&Its Applications,2001,292(1):1-8
20李小川,施明恒.多孔介质通道的分形随机行走描述[C]//传热传质学学术会议.北京:中国工程热物理学会,2009:477-479Li Xiaochuan,Shi Mingheng.Fractal random walk description of porous media channels[C]//Academic Conference on Heat and Mass Transfer.Beijing:Chinese Society of Engineering Thermophysics,2009:477-479
21贝尔.多孔介质流体动力学[M].李竞生,陈崇希,译.北京:中国建筑工业出版社,1983Bear J.Dynamics of fluids in porous media[M].Li Jingsheng,Chen Chongxi,translated.Beijing:China Building Industry Press,1983
22 Chechkin A V,Gorenflo R,Sokolov I M.Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations[J].Physical Review E Statistical Nonlinear&Soft Matter Physics,2002,66:046129
23 Podlubny I.Fractional differential equation[M].San diego,Academic Press,1999
24 Gouyet J F.Physics and fractal structures[M].Berlin:SpringerVerlag,1996
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