Abstract We upscale reactive mixing using effective dispersion coefficients to capture the combined effect of pore-scale heterogeneity and molecular diffusion on the evolution of the mixing interface between two initially segregated dissolved species. Effective dispersion coefficients are defined in terms of the average spatial variance of the solute distribution evolving from a pointlike injection, this means, the transport Green function. We numerically investigate the temporal behavior of the longitudinal effective dispersion coefficients for two porous media of different pore-scale heterogeneity as measured by the statistics of the flow speed, and different Peclet numbers. We find that the effective dispersion coefficients evolve with time, or equivalently travel distance. As the solute samples the pore-scale flow heterogeneity due to advection and transverse diffusion, the effective dispersion coefficients evolve from the value of molecular diffusion to the corresponding hydrodynamic dispersion coefficients. Thus, at times smaller than the diffusion time over a characteristic pore length, the effective dispersion coefficients can be significantly smaller than the hydrodynamic dispersion coefficients. This difference can explain frequently observed mismatches between pore-scale reactive mixing data, and predictions using Darcy scale transport descriptions based on hydrodynamic dispersion coefficients that are constant in time. This suggests that the notion of incomplete mixing on the support scale, can be quantified in terms of effective pore-scale dispersion coefficients. We use effective dispersion in order to approximate the transport Green function in terms of a Gaussian-shaped distribution that is characterized by the effective variance. This is approximation is termed dispersive lamella. Based on this representation, we study reactive mixing between two initially segregated solutes. The dispersive lamella approach accurately predicts the evolution of the product mass of an instantaneous bimolecular reaction obtained from direct numerical simulations. This demonstrates that effective dispersion is an accurate measure for width of the mixing interface between the two reacting species. These results shed some new light on pore-scale mixing, the notion of incomplete mixing, and its prediction and upscaling in terms of an effective mixing model.

中文翻译：

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用于放大孔隙尺度混合和反应的有效分散系数

摘要 我们使用有效分散系数来升级反应混合，以捕捉孔隙尺度异质性和分子扩散对两种最初分离的溶解物种之间混合界面演化的综合影响。有效分散系数是根据从点状注入演化而来的溶质分布的平均空间方差来定义的，这意味着传输格林函数。我们数值研究了两种不同孔隙尺度非均质性多孔介质的纵向有效色散系数的时间行为，如通过流速统计和不同 Peclet 数测量的。我们发现有效色散系数随时间或等价的行进距离而变化。当溶质对由于平流和横向扩散引起的孔隙尺度流动不均匀性进行采样时，有效弥散系数从分子扩散值演变为相应的流体动力学弥散系数。因此，当时间小于特征孔长度上的扩散时间时，有效分散系数可能显着小于流体动力分散系数。这种差异可以解释经常观察到的孔隙尺度反应性混合数据之间的不匹配，以及使用基于时间恒定的流体动力分散系数的达西尺度传输描述的预测。这表明支持尺度上不完全混合的概念可以根据有效孔隙尺度分散系数进行量化。我们使用有效分散来根据以有效方差为特征的高斯分布来近似传输格林函数。这种近似称为色散薄片。基于这种表示，我们研究了两种最初分离的溶质之间的反应性混合。分散薄片方法准确地预测了从直接数值模拟获得的瞬时双分子反应的产物质量的演变。这表明有效分散是两种反应物质之间混合界面宽度的准确度量。这些结果为孔隙尺度混合、不完全混合的概念及其在有效混合模型方面的预测和放大提供了一些新的启示。