Dispersion in fractured media impacts many environmental and geomechanical practices. It is mainly controlled by the structure of fracture networks and the Péclet number $(Pe)$, but predicting it remains challenging. In this study, numerous three-dimensional stochastic discrete fracture networks (DFNs) were generated, where the density, size, and orientation vary significantly. The aperture and conductivity are proportional to the size, following power-laws. Through flow and transport simulation, we evaluated the longitudinal dispersion coefficients $\left(D_L\right)$. We found that, as density increases, the tortuosity decreases and the first passage time distributions approximate bell-shaped curves more closely, which suggests, but does not fully guarantee, that an asymptotic dispersion regime may emerge for denser DFNs, as solute particles traverse more fractures and the macroscopic inter-fracture mixing is more homogeneous. We then determined the $D_L$ values for DFNs in which the time evolution of the variance of particle displacements becomes linear and hence asymptotic. The results show that both $Pe$ and fracture density affect $D_L$, but the former has a much stronger influence than the latter. A new Péclet number $\left(P{e}^c\right)$ was recalculated for all DFNs, where the characteristic length scale accounts for the influence of large fractures. Dimensionless $D_L$ values show a unique power-law relationship with high $P{e}^c$ values. Furthermore, when advection dominates, the dimensionless $D_L$ can be described by a universal finite-size scaling function depending on fracture density and domain sizes. The findings of this study enhance the understanding of transport in fracture networks and imply the potential for predicting $D_L$ in a broad range of scenarios using statistics on fracture parameters obtainable at the field scale.

中文翻译：

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离散断裂网络中 Péclet 数依赖性纵向离散


裂隙介质中的分散会影响许多环境和地质力学实践。它主要受裂隙网络结构和佩克莱特数 $（Pe）$ 的控制，但预测它仍然具有挑战性。在这项研究中，生成了许多三维随机离散断裂网络 （DFN），其中密度、大小和方向变化很大。孔径和电导率与尺寸成正比，遵循幂律。通过流动和传输模拟，我们评估了纵向色散系数 $（D_L）$。我们发现，随着密度的增加，迂曲度减小，第一次通过时间分布更接近钟形曲线，这表明但不能完全保证，对于更密集的 DFN，可能会出现渐近色散状态，因为溶质颗粒穿过更多的裂缝并且宏观裂隙间混合更加均匀。然后，我们确定了 DFN 的 $D_L$ 值，其中粒子位移方差的时间演变变为线性，因此是渐近的。结果表明，$Pe$ 和裂缝密度都对 $D_L$ 产生影响，但前者的影响远大于后者。 为所有 DFN 重新计算了一个新的 Péclet 数 $（P{e}^c）$，其中特征长度尺度解释了大裂缝的影响。无量纲 $D_L$ 值与较高的 P e c$P{e}^{c}$ 值显示出唯一的幂律关系。 ${}^{}$此外，当平流占主导地位时，无量纲${\mathrm{的\; D}}_L$ 可以用通用有限尺寸缩放函数来描述，具体取决于裂缝密度和畴大小。这项研究的结果增强了对裂缝网络中传输的理解，并暗示了使用在野外尺度上获得的裂缝参数统计数据在广泛的情景中预测 $D_L$ 的潜力。