The properties of groundwater aquifers are uncertain at almost all physical locations. Only a small fraction of an aquifer can be sampled and tested. Some statistical model must be chosen and implemented that describes the distribution and correlation structure of the parameters (primarily the hydraulic conductivity) that control the motion of groundwater and dissolved pollutants. Based on the statistical model, replicas of the aquifers are constructed to make predictions of flow and transport. These replicas must contain the structures that may be present across all scales of interest. The replicas should honor the real data where measured (including a possibly non-Gaussian distribution), and should represent correlation that might be restricted to very narrow directional windows; for example, discrete fracture set orientations. The investigators are working on one model that can honor the 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 operator fractional motions are based on the investigators' recent definitions of 3-D fractional integro-differentiation. The operator-fractional motions have what is known in the earth sciences as "generalized scale invariance" in which the index of scaling depends on direction. The investigators are rigorously defining the mathematical properties of these motions and their numerical implementations. They are also studying the correspondence between transport through the operator-fractional aquifer replicas and simple analytic descriptions of transport embodied in fractional-order transport equations and generalized continuous time random walks.
The spreading of pollution in an aquifer is controlled by sediment structures that are present across a huge range of scales. Tiny bits of silt and clay may retain pollutants for years, while buried (remnant) gravelly stream channels might move some pollutants for miles in a short time. To make realistic predictions of drinking water vulnerability or eventual cleanup, it is important to construct models of aquifers that have this kind of multi-scale structure. In another setting more germane to nuclear waste storage, the simultaneous presence of small and large fractures will account for both the slow and fast pathways for the spreading of radioactivity away from future repositories. Representative models of these aquifers must be "buildable" and testable, so the investigators are first pouring a concrete mathematical footing. While the work deals primarily with mathematics and hydrology, the results also contribute to applied studies in physics, fluid dynamics, electrical engineering, and finance. The financial applications are revealed when looking at a graph of a stock price: there are minute-to-minute fluctuations that are similar to year-to-year gains and losses. Some stocks are tightly coupled; others are not. Not all markets respond at the same rate, nor are the magnitudes of the changes easily characterized. Finally, several graduate students and a post-doctoral researcher are also supported by the grant. Each of them is receiving extensive training in both physical sciences and mathematics. This cross-training engenders more fruitful cooperation between the theoretical and applied sciences.
