Bed-load transport is a complex process exhibiting anomalous dynamics, which cannot be efficiently described using the traditional advection–diffusion equation. This study aims at developing and testing a Hausdorff fractal derivative model to characterize scale-dependent, anomalous dynamics of bed-load transport through a heterogeneous gravel-bed. Applications show that the Hausdorff fractal derivative model generally matches the bed sediment snapshots measured in flume experiments with both continuous and instantaneous sediment sources. The order of the Hausdorff fractal derivative is a scale-dependent indicator varying with bed heterogeneity and particle size. For example, bed armoring and size selective transport can cause the fast downstream motion of fine sediment and the enhanced trapping for coarse materials, which can be conveniently quantified by selecting the corresponding order of the Hausdorff fractal derivative in the new model proposed by this study. Further comparison with the fractional derivative model (containing a nonlocal operator to capture long-term memory embedded in both motion and resting of sediment particles) shows that both models can capture anomalous bed-load dynamics, while the Hausdorff fractal derivative model is more attractive due to its local operator and convenient numerical solution.

床层运输是一个表现出异常动力学的复杂过程，无法使用传统的对流扩散方程有效地描述。这项研究的目的是开发和测试Hausdorff分形导数模型，以表征通过非均质砾石床进行的，与规模有关的异常床层输运动力学。应用表明，Hausdorff分形导数模型通常与连续和瞬时沉积物来源的水槽实验中测得的床层沉积物快照相匹配。Hausdorff分形导数的阶是随床非均质性和粒径而变化的尺度相关指标。例如，床身装甲和尺寸选择性运输可导致细颗粒沉积物快速向下游运动，并增强对粗物质的捕集，通过在本研究提出的新模型中选择Hausdorff分形导数的相应阶数，可以方便地对其进行量化。与分数导数模型（包含一个非本地算子以捕获嵌入在沉积物的运动和静止状态中的长期记忆）的进一步比较表明，这两个模型都可以捕获异常的床荷动力学，而Hausdorff分形导数模型由于具有更强的吸引力本地操作员和方便的数值解决方案。