多孔介质中非Fick扩散的分布阶Hausdorff导数扩散模型
Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media
摘要：
大量的理论和实验结果表明，非均质多孔介质中溶质运移具有多尺度行为。为了描述这种非Fickian扩散，这项工作提供了一个分布有序的Hausdorff扩散模型来描述示踪剂在多孔介质中的传输。证明了该模型与扩散系数为非线性时变的扩散方程模型是等价的。结合结构导数，其均方位移（MSD）的示踪剂颗粒被明确地推导出作为双对数函数时的顺序分布的权重函数是一个线性函数的时间导数顺序。该模型可以通过改变权重参数c来捕获加速和减速的反常扩散和超低扩散。在这项研究中，示踪剂在二维欧几里德充满水的孔隙空间中的传输被证明是一个减速子扩散，可以很好地描述的分布顺序Hausdorff扩散模型，c = 1.73。而Hausdorff扩散模型能较好地拟合示踪剂在孔隙固体预分形多孔介质中运移的亚扩散实验数据。
英文摘要：
Many theoretical and experimental results show that solute transport in heterogeneous porous media exhibits multi-scaling behaviors. To describe such non-Fickian diffusions, this work provides a distributed order Hausdorff diffusion model to describe the tracer transport in porous media. This model is proved to be equivalent with the diffusion equation model with a nonlinear time dependent diffusion coefficient. In conjunction with the structural derivative, its mean squared displacement (MSD) of the tracer particles is explicitly derived as a dilogarithm function when the weight function of the order distribution is a linear function of the time derivative order. This model can capture both accelerating and decelerating anomalous and ultraslow diffusions by varying the weight parameter c. In this study, the tracer transport in water-filled pore spaces of two-dimensional Euclidean is demonstrated as a decelerating sub-diffusion, and can well be described by the distributed order Hausdorff diffusion model with c = 1.73. While the Hausdorff diffusion model can accurately fit the sub-diffusion experimental data of the tracer transport in the pore-solid prefractal porous media.
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