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

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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.