Contaminant transport through natural aquifers typically exhibits pre-asymptotic or transient anomalous behavior on the space and time scale critical to most environmental concerns. Complex and usually unpredictable medium heterogeneity at all relevant scales motivates the application of non-local transport theories. The proposed work will develop tempered stable models, which generalize standard non-local transport theories by adjusting fractal power-laws, to simulate pre-asymptotic transport and reveal the nature of real-world dispersion missed previously. There will be three major outcomes, including (1) a novel non-local transport theory and model based on tempered power laws that can efficiently simulate transient anomalous diffusion, (2) a quantitative linkage between the observable statistics of natural heterogeneous media and the model parameters built by a systematic Monte Carlo study, and (3) a convenient software suite with open source codes that solve and apply the model. This collaborative research will also test the model, the solver and the model predictability, by using historical tracer data and well-studied aquifer information. A careful consideration of the physical meaning of model components, and connections to statistical aquifer properties, will ensure that the resulting model is not limited to curve fitting applications.
Accurate prediction of contaminant migration in real-world aquifers is critical to groundwater protection and cleanup. The proposed work will develop appropriate transport theory and build effective modeling components to address this problem. Hence this research is both highly theoretical and applied. In particular, the proposed work more accurately represents the underlying link between fractional calculus and power-law statistics in real aquifer material. The PI team includes mathematicians and hydrologists, forming interdisciplinary cooperation in cutting edge research.
This project developed new models for the transport of pollutants in water, by developing a new kind of calculus. This has allowed us to develop superior models for pollution migration in rivers and underground streams. The new tempered fractional calculus interoplates between traditional models based on Newton's calculus, and fractional calculus models developed during the past few decades. Newtonian calculus describes the behavior of a traditional random walk, where a contaminant particle takes a series of steps in a random direction. Fractional calculus allows very long steps, and the probabilitty of stepping further than a fixed distance falls off like a power of that distance. This creates interesting fractal paths, and provides a nice model for many applications. Our research extends this model, where the power law jump probability is multiplied by an exponential tempering factor. The accompanying figure shows the effect of tempering on a particle path. As the tempering parameter gets larger, the path transitions from fractional (with jumps) to Newtonian (no jumps). Another part of this project used the same ideas to develop a tempered fractional Brownian motion, as a model for tempered fractional random walks. The tempered fractional Brownian motion interpolates between the Newtonian process of Brownian motion, which describes the long-time behavior of a Newtonian random walk, and fractional Brownian motion, which captures the behavior of a fractional (power law) random walk after a long time has passed. We showed that the steps of a tempered fractional Brownian motion, called tempered fractional Gaussian noise, provide the first concrete model that reproduces the Davenport spectrum for wind speed. Thus, tempered fractional Brownian motion can be used to effectively simulate wind gusting near the earth surface, for design of buildings, antennae, and windmills. We have also developed a related time series model called ARTFIMA, for "autoregressive tempered fractionally integrated moving average," that extends the successful ARFIMA model. We have applied this ARTFIMA model to geophysical turbulence, and have shown that it gives a better fit than other existing models. Hence the ARTFIMA model can be used to faithfully simulate water velocities in lakes and rivers.
