Abstract While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model.

中文翻译：

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相关速度场中非线性反应输运的放大

摘要 虽然通常的非局部输运模型已被证明可以成功地再现由异质含水层输运产生的突破曲线，但悬而未决的问题包括放大的控制方程和次尺度异质性之间的正式联系，以及解释影响的能力在存在非线性多组分反应的情况下，有效化学反应速率的异质性。基于速度马尔可夫模型的时域随机游走方法通过将溶质粒子沿其轨迹的连续位置的速度的空间相关性结合起来，提供了解决这些问题的框架。这些方法通常依赖于粒子跟踪方法，然而，这在计算上可能很繁重，尤其是在寻求对非线性反应进行建模时。在本文中，提出了一个积分微分方程，用于在依赖于表示速度相关结构的联结器的异构介质中放大多分量反应传输。为此，我们将溶质的浓度或通量表示为其速度的分布。然后我们推导出积分微分方程，该方程控制浓度分布在定义为速度等级的数量上的演变。通过这种方式，基于遍历横截面速度分布和参数化 copula 函数来预测远离源的突破曲线的空间演化，该函数表示沿其轨迹的溶质粒子的速度等级之间的相关性。