A fractal advection-dispersion equation and a fractional space-time advection-dispersion equation have been developed to improve the simulation of groundwater transport in fractured aquifers. The space-time fractional advection-dispersion simulation is limited due to complex algorithms and the computational power required; conversely, the fractal advection-dispersion equation can be solved simply, yet only considers the fractal derivative in space. These limitations lead to combining these methods, creating a fractional and fractal advection-dispersion equation to provide an efficient non-local, in both space and time, modeling tool. The fractional and fractal model has two parameters, fractional order ($ \alpha $) and fractal dimension ($ \beta $), where simulations are valid for specific combinations. The range of valid combinations reduces with decreasing fractional order and fractal dimension, and a final recommendation of $ \; 0.7 \leq \alpha, \beta \leq 1 $ is made. The fractional and fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model.

中文翻译：

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分形和分形对流-弥散模型

分形平流-弥散方程和分数时空平流-弥散方程已被开发，以改进对裂缝含水层中地下水输送的模拟。由于复杂的算法和所需的计算能力，时空分数对流扩散模拟受到限制；反之，分形对流-色散方程可以简单求解，但只考虑空间中的分形导数。这些限制导致结合这些方法，创建分数和分形对流-弥散方程，以提供有效的非局部空间和时间建模工具。分数和分形模型有两个参数，分数阶 ($\alpha $) 和分形维数 ($\beta $)，其中模拟对特定组合有效。有效组合的范围随着分数阶和分形维数的减少而减少，最终建议为 $\; 0.7 \leq \alpha, \beta \leq 1 $ 制成。分数和分形模型提供了一种灵活的工具来模拟异常扩散，其中分数阶控制突破曲线峰，分形维数控制峰的位置和拖尾效应。这两个控件可能提供工具来改进经典模型无法描述的异常突破曲线的表示。分形维数控制着峰值和拖尾效应的位置。这两个控件可能提供工具来改进经典模型无法描述的异常突破曲线的表示。分形维数控制着峰值和拖尾效应的位置。这两个控件可能提供工具来改进经典模型无法描述的异常突破曲线的表示。