We propose a theoretical framework to calculate from first principles the distribution of arrival times, W(t), of non-sorbing solute due to advection in two- and three-dimensional heterogeneous porous media and then compare these theoretical predictions with experimental values. Using the related spatial concentration distribution at a given time, the PIs will then predict the time dependence of the longitudinal dispersion coefficient, Dl(t). The theory, based partly on critical path analysis (using cluster statistics of percolation) and partly on percolation scaling of tortuosity, links existing work of one of the PIs in calculating a distribution of rate-limiting conductances with published work of the Eugene Stanley group at Boston University regarding the scaling of the typical travel time with length. We compare our predictions with particle tracking simulations of flow performed by other researchers on a two-dimensional percolation structure, for which the Navier-Stokes equations were solved numerically. Our results compare very favorably with the numerical simulations. Using parameters appropriate for a very narrow pore size distribution (rmin/rmax=0.7), we are able to fit their data at an early time and, using no new parameters, predict W(t) in a system five times larger. The theoretical results are compatible with experiment as well. We propose to test the theoretical work rigorously on experiments and give hypotheses to guide the tests. The experimental results that we address have been the subject of decades of concentrated study without resolution. Moreover, the results of the simulations have not yet been addressed by anyone. Since traditional modeling of dispersion in porous media yields Gaussian spreading in the direction of transport, and since our method is compatible with the much more commonly observed power-law spreading (Margolin and Berkowitz, 2000), it has the potential to transform the way dispersion is modeled. Understanding dispersion is relevant for modeling radioactive tracers to determine groundwater ages or histories of migrating crustal fluids (and thus constraining models), spreading of contaminant plumes, and for agricultural purposes. Our treatment, which excludes molecular diffusion, is not relevant for flows with very small Peclet numbers, meaning that including effects of diffusion is probably the most important extension we can envision. By carrying out this project we will be providing research opportunities for undergraduate students in the earth and environmental sciences department, acquainting them with the most current status of research in hydrogeology. We will also introduce the students to modern quantitative methods in probability theory
