Incorporation of effects of diffusion on longitudinal dispersion by advection

分散是指在多孔介质中夹带的颗粒在时间和空间上扩散的过程。这种扩散很少是高斯分布，具有更粗的（幂律）尾部，并且不能使用传统的偏微分方程来预测。我们建立在一个现有的程序，利用事先计算的粒子速度和路径长度的分布预测溶质到达时间的分布忽略分子扩散。计算是基于融合的渗流理论技术：集群统计的渗流理论，关键路径分析，和路径曲折。每个流动路径通过该路径上的最小（速率限制）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的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，并且幂律行为在长距离处逐渐变为高斯行为（如已知的对于单个毛细管发生的那样）。扩散相对于流动的影响是使用称为佩克莱数Pe的量来估计的。我们发现，高斯行为集在一个距离，这是成比例的功率Pe。我们建议的研究包括：1）生成颗粒在孔i处扩散离开流动路径的概率fi，2）使用fi来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代码中，3）将任意空间位置x处的到达时间W（t）的所得分布与任意时间t处的空间溶质分布W（x）相关联，4）计算溶质空间分布的矩。了解溶质浓度的时空演变在监测地下溢油、评估地下水年龄分布、指导植物吸收矿物和肥料以及设计强化采油工艺等方面具有重要意义。因此，这项研究可以加强对农业，石油和采矿业，有毒废物清理和风险评估，地质测序，地下水年龄和来源估计有关的过程的理解。我们的工作的第二个潜在影响是，它可能导致人们在这一研究领域的观点发生变化，重新强调连续介质力学和相关微分方程以外的技术。

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.