We present an approach for quantifying displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanism. It shows that the evolution of dispersion can be captured as the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the short time at which transported particles start exploring the flow variability and the large time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local-scale dispersion as an additional decorrelation mechanism. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media with possible applications, for example, in groundwater contamination modelling, underground gas storage, and geothermal energy production.

中文翻译：

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非均质多孔介质中位移前沿演化的时间相关色散系数


我们提出了一种基于时间相关的表观色散系数概念来量化非均质多孔介质中位移前沿的方法。恒定渐近宏观色散的概念通常会高估位移前沿扫过的面积，并导致不切实际的上游色散。我们证明，大范围的锋面扩散可以通过一维平流-色散方程来捕获，该方程由适当选择的随时间演变的色散系数参数化。对于纯粹的平流锋面扩散，我们基于预测连续时间随机游走方法推导了解析表达式，该方法适用于高度异质多孔介质。该分析阐明了溶质传播时间的可变性作为关键的纵向扩散机制。它表明色散的演变可以被捕获为在两个主要时间尺度上衰减的指数之和。在基于粒子的图像中，这些尺度标志着传输粒子开始探索流动变化的短时间和最慢粒子开始与其传输速度去相关的长时间。基于这些见解，我们提出了一个启发式公式，该公式将局部尺度色散的影响解释为额外的去相关机制。纵向分散系数的启发式表达式捕获了大范围佩克莱特数和异质性方差的溶质扩散。所提出的方法通过直接数值模拟进行了测试。 它提供了一种稳健而快速的方法来量化非均质多孔介质中位移前沿的演变，并具有可能的应用，例如地下水污染建模、地下天然气储存和地热能生产。