HR: 0800h
AN: H11D-1311    [Abstracts]
TI: Random-walk Solutions of Fractional Advection-dispersion Equations
AU: * Zhang, Y
EM: yong.zhang@dri.edu
AF: Division of Hydrologic Sciences, Desert Research Institute, 2215 Raggio Parway, Reno, NV 89512 United States
AU: Benson, D A
EM: dbenson@mines.edu
AF: Department of Geology and Geological Engineering, Colorado School of Mines, 1516 Illinois Street, Golden, CO 80401 United States
AU: Meerschaert, M M
EM: mcubed@maths.otago.ac.nz
AF: Department of Mathematics and Statistics, University of Otago, Dunedin, PO Box 56 New Zealand
AU: Scheffler, H
EM: pscheff@unr.edu
AF: Department of Mathematics, University of Nevada, 1664 North Virginia St, Reno, NV 89557 United States
AB: The spatially fractional advection-dispersion equation (fADE) broadens the applicability of the traditional second-order advection-dispersion equation (ADE) for contaminant transport by capturing the anomalous dispersion observed frequently in laboratory and field experiments. The equation is nonlocal in character, since it describes the spread of solute mass over large distances via a convolutional fractional derivative. The characteristic coefficients - velocity and dispersion tensor - may also vary locally in space, so that the strength of the nonlocal spreading may be a function of the local-scale subsurface heterogeneity. Numerical methods are required to solve the fADE with space-dependent parameters since analytical solutions are unavailable. Similar to the solution of the second-order ADE, a random walk may be the method of choice for simulating anomalous transport through large flow systems in heterogeneous porous media or fracture networks. To develop the random walk to solve the fADE, we first explore the forms of fADE with space-dependent parameters using a generalized mass conservation law. Different fADEs arise depending on the non-locality of fluxes and the scale index representing the degree of non-locality. Then we solve each fADE explicitly by building the appropriate Langevin equation after the relationship between the non-linear Langevin equation and the generalized Fokker-Planck equation is established by the adjoint method. Additional techniques are also required when applying the random walk method to anomalous dispersion of solutes across abrupt interfaces of depositional materials where the dispersion coefficients may be discontinuous. The current techniques for resolving the problem, including the interpolation scheme, the reflection principle, and LaBolle's stochastic differential equation method, may be no longer effective in this case due to the skewness and the heavy tail of Lévy motions, which are solutions to the fractional diffusion equations. Therefore, novel methods, such as considering the different jumping probabilities of particles along up- and down-stream directions are needed for solving the fADEs with discontinuous coefficients.
DE: 1829 Groundwater hydrology
DE: 1832 Groundwater transport
SC: Hydrology [H]
MN: Fall Meeting 2005