Collaborative Research: CMG: Multi-scaling Random Fields and Pollution Migration

地下水含水层的性质在几乎所有物理位置都是不确定的。只有一小部分含水层可以取样和测试。必须选择和实施一些统计模型来描述控制地下水和溶解污染物运动的参数(主要是渗透系数)的分布和相关结构。在统计模型的基础上，建立了含水层的复制品，以预测流量和运输。这些复制品必须包含可能存在于所有感兴趣的范围内的结构。复制品应遵循测量的真实数据(可能包括非高斯分布)，并应表示可能被限制在非常窄的定向窗口内的相关性；例如，离散的裂缝设置。研究人员正在研究一种模型，该模型可以模拟许多经过充分研究的含水层(颗粒状和/或碎裂状)的数据。该模型使用算子分数噪声和运动，因为含水层的尺度或分维特性随方向变化。算符分数运动基于研究人员最近对三维分数积分微分学的定义。算符分数运动具有地球科学中所熟知的“广义标度不变性”，其中标度指数取决于方向。研究人员正在严格定义这些运动的数学性质及其数值实现。他们还在研究通过算子分数含水层复制品的运移与分数阶运移方程和广义连续时间随机游动中体现的运移的简单解析描述之间的对应关系。含水层中污染的扩散受沉积结构的控制，这些结构存在于大范围的尺度上。细小的泥沙和粘土可能会将污染物滞留数年，而埋藏的(残留的)砾石河道可能会在短时间内将一些污染物移动数英里。为了对饮用水脆弱性或最终清除做出现实的预测，建立具有这种多尺度结构的含水层模型是很重要的。在另一种更接近于核废料储存的环境中，同时存在的小的和大的裂缝将解释放射性物质从未来的储存库扩散出去的慢速和快速路径。这些含水层的典型模型必须是“可建造的”和可测试的，所以调查人员首先要浇筑一个具体的数学基础。虽然这项工作主要涉及数学和水文学，但其成果也有助于物理、流体动力学、电气工程和金融方面的应用研究。当查看股票价格的图表时，财务应用就会被揭示出来：每分钟的波动类似于年度收益和亏损。一些股票紧密相连，而另一些则不是。没有哪个市场会以同样的速度做出反应，变化的幅度也不容易确定。最后，几名研究生和博士后研究员也得到了这笔拨款的支持。他们每个人都在接受物理和数学方面的广泛训练。这种交叉培训在理论科学和应用科学之间产生了更富有成效的合作。

The properties of groundwater aquifers are uncertain at almost allphysical locations. Only a small fraction of an aquifer can be sampledand tested. Some statistical model must be chosen and implemented thatdescribes the distribution and correlation structure of the parameters(primarily the hydraulic conductivity) that control the motion ofgroundwater and dissolved pollutants. Based on the statistical model,replicas of the aquifers are constructed to make predictions of flow andtransport. These replicas must contain the structures that may bepresent across all scales of interest. The replicas should honor thereal data where measured (including a possibly non-Gaussiandistribution), and should represent correlation that might be restrictedto very narrow directional windows; for example, discrete fracture setorientations. The investigators are working on one model that can honorthe data from many well-studied aquifers (granular and/or fractured). The model uses operator-fractional noise and motions, since the scaling,or fractal, properties of aquifers vary with direction. The operatorfractional motions are based on the investigators' recent definitions of3-D fractional integro-differentiation. The operator-fractional motionshave what is known in the earth sciences as "generalized scaleinvariance" in which the index of scaling depends on direction. Theinvestigators are rigorously defining the mathematical properties ofthese motions and their numerical implementations. They are alsostudying the correspondence between transport through theoperator-fractional aquifer replicas and simple analytic descriptions oftransport embodied in fractional-order transport equations andgeneralized continuous time random walks. The spreading of pollution in an aquifer is controlled by sedimentstructures that are present across a huge range of scales. Tiny bits ofsilt and clay may retain pollutants for years, while buried (remnant)gravelly stream channels might move some pollutants for miles in a shorttime. To make realistic predictions of drinking water vulnerability oreventual cleanup, it is important to construct models of aquifers thathave this kind of multi-scale structure. In another setting moregermane to nuclear waste storage, the simultaneous presence of small andlarge fractures will account for both the slow and fast pathways for thespreading of radioactivity away from future repositories. Representative models of these aquifers must be "buildable" andtestable, so the investigators are first pouring a concrete mathematicalfooting. While the work deals primarily with mathematics and hydrology,the results also contribute to applied studies in physics, fluiddynamics, electrical engineering, and finance. The financialapplications are revealed when looking at a graph of a stock price:there are minute-to-minute fluctuations that are similar to year-to-yeargains and losses. Some stocks are tightly coupled; others are not. Notall markets respond at the same rate, nor are the magnitudes of thechanges easily characterized. Finally, several graduate students and apost-doctoral researcher are also supported by the grant. Each of themis receiving extensive training in both physical sciences andmathematics. This cross-training engenders more fruitful cooperationbetween the theoretical and applied sciences.