Quantitative predictions of scalar transport in natural porous media is a nontrivial task given the presence of multi-scale spatial heterogeneity in the permeability field. Due to data scarcity, the structural map of the permeability field is subject to uncertainty and therefore, model predictions are uncertain. For such reasons, probabilistic models of flow and transport in natural porous media are required in risk assessment and to provide reliable decision making under uncertainty. Further complexities arise when the viscosity of the injected solute differs from that of the ambient fluid. Under the presence of viscosity contrast, hydrodynamic instabilities give rise to viscous fingering, which induces additional disorder in both velocity and solute concentration fields. This work examines the combined role of viscous fingering and permeability heterogeneity in the probabilistic description of transport predictions. In particular, we focus on metrics that are important for risk analysis, such as the solute plume’s early arrival times and the maximum concentration observed at a given location. We propose to use the Projection Pursuit Adaptation (PPA) method in the Polynomial Chaos Expansion (PCE) framework to quantify uncertainty in transport model predictions. The PPA method is a data-driven approach that optimally represents a given quantity of interest in a low-dimensional manifold. Unlike other dimension reduction techniques in uncertainty quantification, the PPA method utilizes non-linear information of the quantity of interest to identify the low-dimensional manifold, thereby increasing the likelihood of finding a more accurate lower-dimensional space. Moreover, the PPA model converges to the physical solution in a mean squared sense with respect to the polynomial order, enabling the construction of a converged model even with limited available data. Then, the PPA results are compared to Monte Carlo simulations using the same amount of data. This comparison illustrates that while Monte Carlo simulations are able to capture low-order statistics, they struggle to represent more detailed probability density functions. Our results show how the combined effect of permeability heterogeneity and viscosity contrast can enhance the mobility of the solute plume.

中文翻译：

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非均质多孔介质中流体动力学不稳定流动下标量输运的概率评估

鉴于渗透率场中存在多尺度空间异质性，天然多孔介质中标量传输的定量预测是一项艰巨的任务。由于数据稀缺，渗透率场的结构图存在不确定性，因此模型预测具有不确定性。因此，在风险评估中需要天然多孔介质中流动和传输的概率模型，并在不确定性下提供可靠的决策。当注入溶质的粘度与环境流体的粘度不同时，会出现进一步的复杂性。在存在粘度对比的情况下，流体动力学不稳定性会引起粘性指进，从而在速度和溶质浓度场中引起额外的紊乱。这项工作研究了粘性指进和渗透率异质性在输运预测的概率描述中的综合作用。我们特别关注对风险分析很重要的指标，例如溶质羽流的提前到达时间和在给定位置观察到的最大浓度。我们建议在多项式混沌展开（PCE）框架中使用投影追踪适应（PPA）方法来量化传输模型预测中的不确定性。 PPA 方法是一种数据驱动的方法，可以最佳地表示低维流形中给定的感兴趣数量。与不确定性量化中的其他降维技术不同，PPA方法利用感兴趣量的非线性信息来识别低维流形，从而增加找到更准确的低维空间的可能性。此外，PPA 模型在多项式阶数的均方意义上收敛到物理解，即使可用数据有限也能构建收敛模型。然后，使用相同量的数据将 PPA 结果与蒙特卡罗模拟进行比较。这一比较表明，虽然蒙特卡罗模拟能够捕获低阶统计数据，但它们很难表示更详细的概率密度函数。我们的结果表明渗透率异质性和粘度对比的综合作用如何增强溶质羽流的流动性。