Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications
Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications
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DOI：
10.1016/j.advwatres.2009.01.008
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发表时间：
2009-04
影响因子：
4.7
通讯作者：
Yong Zhang;D. Benson;D. Reeves
中科院分区：
文献类型：
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作者：
Yong Zhang;D. Benson;D. Reeves
我们调查的时空非局部性分数阶导数模型作为一个可能的解释区域尺度的异常色散与重尾。系统地分析和比较了4种分数阶对流-弥散方程（fADE）模式的性质，包括具有最大正偏度或最大负偏度的空间fADE、具有时间分数阶导数0<γ<1的时间fADE和具有1<γ<2的时间fADE的扩展。空间fADE描述了局部浓度变化对大范围空间区域的依赖性（即，空间非定域性），而时间fADE描述移动的和多个非移动的相之间的动态质量交换并因此记录浓度“加载”的时间历史（即，时间非局部性（Time Nonlocality）。然后，我们应用的fADES作为模型的异常分散到四个广泛研究，区域尺度，自然系统，包括一个山坡组成的断裂土壤，河流同时活跃的流动区和各种死区，一个相对均匀的冰川河流含水层为主的层状砂和砾石，和一个高度异质性的冲积含水层包含优先的流动路径和丰富的弱透水层。我们发现，在每个站点观察到的异常色散可能没有合理或充分的特点，以前的研究。特别是，使用的空间fADE小于最大正偏度意味着空间依赖于下游浓度，可能是不现实的物理溶质输运在流域集水区和河流（死区对溶质输运的影响可以描述的时间，而不是空间，分数模型）。场尺度的输运研究表明，大范围的溶质位移可以描述的空间非本地，分数阶导数模型，和长的等待时间可以有效地描述的时间非本地，分数阶模型。非局部参数和异质性之间的未知的定量关系，以及浓度分布的相似性，是不同的非局部传输模型的解决方案，都表明区分代表性的非局部性（时间和/或空间）的重要性，为任何给定的区域尺度的异常扩散过程。
We investigate the spatiotemporal nonlocality underlying fractional-derivative models as a possible explanation for regional-scale anomalous dispersion with heavy tails. Properties of four fractional-order advection–dispersion equation (fADE) models were analyzed and compared systematically, including the space fADEs with either maximally positive or negative skewness, the time fADE with a temporal fractional-derivative 0<γ<1, and the extension of the time fADE with 1<γ<2. Space fADEs describe the dependence of local concentration change on a wide range of spatial zones (i.e., the space nonlocality), while time fADEs describe dynamic mass exchange between mobile and multiple immobile phases and therefore record the temporal history of concentration “loading” (i.e., the time-nonlocality). We then applied the fADEs as models of anomalous dispersion to four extensively-studied, regional-scale, natural systems, including a hillslope composed of fractured soils, a river with simultaneous active flow zones and various dead-zones, a relatively homogeneous glaciofluvial aquifer dominated by stratified sand and gravel, and a highly heterogeneous alluvial aquifer containing both preferential flowpaths and abundant aquitards. We find that the anomalous dispersion observed at each site might not be characterized reasonably or sufficiently by previous studies. In particular, the use of the space fADE with less than maximally positive skewness implies a spatial dependence on downstream concentrations that may not be physically realistic for solute transport in watershed catchments and rivers (where the influence of dead-zones on solute transport can be described by a temporal, not spatial, fractional model). Field-scale transport studies show that large ranges of solute displacement can be described by a space nonlocal, fractional-derivative model, and long waiting times can be described efficiently by a time-nonlocal, fractional model. The unknown quantitative relationship between the nonlocal parameters and the heterogeneity, and the similarity in concentration profiles that are solutions to the different nonlocal transport models, all demonstrate the importance of distinguishing the representative nonlocality (time and/or space) for any given regional-scale anomalous dispersion process.