Garrard Rhiannon M, Zhang Yong, Wei Song, Sun HongGuang, Qian Jiazhong
Department of Geological Sciences, University of Alabama, Tuscaloosa, AL, 35406.
College of Mechanics and Materials, Hohai University, Jiangsu Province, 211100, China.
Ground Water. 2017 Nov;55(6):857-870. doi: 10.1111/gwat.12532. Epub 2017 Jul 10.
Time nonlocal transport models such as the time fractional advection-dispersion equation (t-fADE) were proposed to capture well-documented non-Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non-Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t-fADE model. Fitting exercises show that the effective dispersion coefficient in the t-fADE, although differing subtly from the dispersion coefficient in the standard advection-dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t-fADE, the motion-independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale-dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t-fADE with a constant dispersion coefficient cannot capture scale-dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real-world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.
诸如时间分数阶对流弥散方程（t-fADE）之类的时间非局部输运模型被提出来，用于描述非均质介质中保守溶质输运时充分记录的非菲克动力学，其潜在假设是物理模型中嵌入的时间非局部性（即当前浓度变化受先前浓度负荷影响）可以使有效弥散系数摆脱尺度依赖性。然而，这一假设从未使用实际数据进行过系统检验。本研究通过使用有前景的 tempered t-fADE 模型，捕捉文献（Huang 等人，1995 年）中记录的以及在我们实验室从小尺度到中等尺度观测到的非菲克输运（可能是由于溶质滞留），填补了这一历史知识空白。拟合结果表明，t-fADE 中的有效弥散系数虽然与标准对流弥散方程中的弥散系数略有不同，但对于非均质和宏观均质砂柱，其均随运移距离（从 0.5 米到 12 米不等）呈非线性增加。进一步分析表明，虽然 t-fADE 中的时间非局部参数可以有效捕捉相对静止区域中的溶质滞留，但流动区域中与运动无关的溶质运移受主体介质中局部速度的空间演化影响，导致弥散系数具有尺度依赖性。对于其他将溶质滞留和跳跃（即位移）分开的标准时间非局部输运模型，也会得到相同的结果。因此，具有恒定弥散系数的 t-fADE 无法捕捉饱和多孔介质中尺度依赖性弥散，这对随机水文地质方法在量化实际的渐近前输运中的应用提出了挑战。因此，有必要使用例如新颖的从属方法对时间非局部模型进行改进，以纳入局部速度的空间演化，而无需添加繁琐的参数。
