Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

被引:299
作者
Zhang, Yong [1 ]
Benson, David A. [2 ]
Reeves, Donald M. [3 ]
机构
[1] Desert Res Inst, Div Hydrol Sci, Las Vegas, NV 89119 USA
[2] Colorado Sch Mines, Dept Geol & Geol Engn, Golden, CO 80401 USA
[3] Desert Res Inst, Div Hydrol Sci, Reno, NV 89512 USA
基金
美国国家科学基金会;
关键词
Anomalous dispersion; Space nonlocality; Time-nonlocality; Fractional ADE; Field application; ADVECTION-DISPERSION EQUATION; HETEROGENEOUS POROUS-MEDIA; DUAL-POROSITY MODEL; FORCED-GRADIENT EXPERIMENT; SOLUTE TRANSPORT; RANDOM-WALKS; MASS-TRANSFER; HYDRAULIC CONDUCTIVITY; ANOMALOUS DIFFUSION; TRACER TEST;
D O I
10.1016/j.advwatres.2009.01.008
中图分类号
TV21 [水资源调查与水利规划];
学科分类号
081501 ;
摘要
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 < gamma < 1, and the extension of the time fADE with 1 < gamma < 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. (C) 2009 Elsevier Ltd. All rights reserved.
引用
收藏
页码:561 / 581
页数:21
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