This work presents the first attempt to develop unconditionally stable, implicit finite difference solutions of one-sided spatial fractional advection-dispersion equation (s-FADE) by imposing the nonzero Dirichlet boundary condition (ND BC) or the nonzero fractional Robin boundary condition (NFR BC) at inlet boundary and the zero fractional Neumann boundary condition (ZFN BC) at outlet boundary. The results of the numerical studies performed using artificial solute transport parameters demonstrated that the numerical solution with the NFR BC as the inlet boundary produced much more realistic concentration values. The numerical solution with the NFR BC at the inlet boundary was capable of correctly describing the Fickian and non-Fickian behaviors of the solute transport at different α values, and it had the relatively same accuracy at different numbers of the spatial nodes. Also, the practical application of the numerical solution with the NFR BC as the inlet boundary was investigated by conducting tracer experiments in homogeneous and heterogeneous soil columns. According to the obtained results, this numerical solution described well solute transport in the homogenous and heterogeneous soils. The α values of the homogeneous and heterogeneous soils were obtained in the ranges of 1.849–1.999 and 1.248–1.570, respectively, which were in excellent agreement with the physical properties of the soils. In a nutshell, the numerical solution of the s-FADE with the NFR BC as the inlet boundary can be successfully applied to describe the solute transport in the homogeneous and heterogeneous soils with bounded spatial domains.

中文翻译：

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有界域饱和多孔介质中溶质运移的数值模拟

这项工作首次尝试通过施加非零狄利克雷边界条件 (ND BC) 或非零分数罗宾边界条件 (NFR) 来开发单边空间分数阶对流-色散方程 (s-FADE) 的无条件稳定、隐式有限差分解BC) 在入口边界和零分数 Neumann 边界条件 (ZFN BC) 在出口边界。使用人工溶质输运参数进行的数值研究结果表明，以 NFR BC 作为入口边界的数值解产生了更现实的浓度值。NFR BC 在入口边界处的数值解能够正确描述不同 α 值下溶质输运的 Fickian 和非 Fickian 行为，并且在不同数量的空间节点上具有相对相同的精度。此外，通过在均质和非均质土柱中进行示踪实验，研究了以 NFR BC 作为入口边界的数值解的实际应用。根据所得结果，该数值解很好地描述了均质和非均质土壤中的溶质运移。均质和异质土壤的 α 值分别在 1.849-1.999 和 1.248-1.570 的范围内获得，这与土壤的物理性质非常吻合。简而言之，以 NFR BC 作为入口边界的 s-FADE 数值解可以成功地应用于描述具有有界空间域的均质和非均质土壤中的溶质输运。