Incorporation of effects of diffusion on longitudinal dispersion by advection

弥散是携带在流动中的颗粒在多孔介质中在时间和空间中扩散的过程。这种扩散很少是高斯型的，尾巴更粗(幂定律)，不能用传统的偏微分方程式来预测。我们建立在现有程序的基础上，该程序利用粒子速度和路径长度分布的先前计算来预测溶质到达时间的分布，而忽略了分子扩散。这些计算基于渗流理论技术的融合：渗流理论的集群统计、关键路径分析和路径曲折。每条流动路径通过该路径上的最小(限速)电导来定义。到达时间分布中的长时间尾部与渗流阈值附近颗粒传输的曲折路径有关。然而，当包括扩散时，颗粒不能在这些路径上停留如此长的时间，并且在长距离处的幂定律行为分级为高斯行为(就像已知的单个毛细管所发生的那样)。扩散相对于流动的影响是使用一个称为Peclet数Pe的量来估计的。我们发现，高斯行为在与Pe次方成正比的距离处开始。我们建议的研究包括：1)生成颗粒在孔i处扩散出流动路径的概率FI，2)使用FI生成颗粒在特定流动路径上停留任意长度的概率并将其并入现有的代码，3)将所得到的到达时间分布W(T)在任意空间位置x与空间溶质分布W(X)相关联，t，4)计算空间溶质分布的矩。了解溶质浓度的时空演变对于监测地下溢出、评估地下水年龄分布、指导植物吸收矿物和化肥以及设计强化采油工艺非常重要。因此，这项研究可以增进对与农业、石油和采矿业、有毒废物清理和风险评估、地质排序以及地下水年龄和来源估计有关的过程的了解。我们工作的第二个潜在影响是，它可能导致这一研究领域的人的观点发生变化，重新强调连续介质力学和相关微分方程式以外的技术。

Dispersion is the process by which particles entrained in flow through a porous medium spread in time and space. This spreading is seldom Gaussian, having much fatter (power-law) tails, and cannot be predicted using conventional partial differential equations. We build on an existing procedure that utilized prior calculations of the distribution of particle velocities and path lengths to predict the distribution of solute arrival times neglecting molecular diffusion. The calculations were based on a fusion of percolation theoretical techniques: cluster statistics of percolation theory, critical path analysis, and path tortuosity. Each flow path is defined through the smallest (rate-limiting) conductance on that path. The long-time tail in the distribution of arrival times relates to those tortuous paths of particle transport near the percolation threshold. When diffusion is included, however, particles cannot stay on these paths for such long times and power law behavior grades into Gaussian behavior at long distances (as is already known to occur for an individual capillary). Effects of diffusion relative to flow are estimated using a quantity called the Peclet number, Pe. We find that Gaussian behavior sets on at a distance which is proportional to a power of Pe. Our proposed research includes: 1) Generating the probability, fi, that a particle diffuses off a flow path at pore i, 2) Using fi to generate the probability that a particle stays on a particular flow path to an arbitrary length and incorporate into existing code, 3) Relating the resulting distribution of arrival times, W(t), at arbitrary spatial position, x, to a spatial solute distribution, W(x) at arbitrary time, t, 4) Calculating the moments of the spatial solute distribution. Understanding spatio-temporal evolution of the concentrations of solutes is important in monitoring subsurface spills, evaluating groundwater age distributions, guiding plant uptake of minerals and fertilizers, and design of enhanced oil-recovery processes. Thus this research can enhance understanding of processes relevant to agriculture, oil and mining industries, toxic waste clean up and risk assessment, geological sequencing, and groundwater age and source estimation. The second potential impact of our work is that it could lead to a change in perspective of the people in this field of research, with renewed emphasis on techniques other than those of continuum mechanics and the associated differential equations.