HR: 0800h
AN: H51F-0425    [Abstracts]
TI: A Lagrangian Approach to Characterizing Dispersion in Heterogeneous Porous Media
AU: * Dean, D W
EM: ddean@math.cudenver.edu
AF: University Of Colorado At Denver, Department Of Mathematics 1250 14th Street P.O. Box 173364 Campus Box 170, Denver, CO 80217-3364 United States
AB: In the Lagrangian framework, transport is developed in terms of homogeneous indivisible fluid particles. The trajectories of the fluid particles are interpreted mathematically as sample paths of a stochastic process. From its origin, the total displacement of the particle consists of a convection component and a dispersion component. The dispersion component turns out to be equal to one-half the time derivative of the displacement covariance tensor. In keeping with the Lagrangian point of view, we are developing particle tracking models for subsurface transport in porous media. The models are for one and two phase flow and are based on stochastic calculus methods. In the single-phase case, a Lagrangian framework for the dispersion tensor is presented that allows assumptions to be local. We start with the displacement covariance tensor and use Darcy's law to derive our numerical upscaling formula for dispersion that uses only assumptions local to the computational grid-blocks. A local dispersion formulation is given for individual grid-blocks and then extended to a global one through a concatenation of local grid-block calculations along trajectories. This implementation can be thought of as a numerical approximation of the time integral of the velocity covariances along the particle trajectories, which in turn is the time derivative of the displacement covariance tensor. A simplification of the result is available by assuming a steady-state flow. In the two-phase case, we attempt to apply the method used for single-phase flow to the Darcy law for the non-wetting or NAPL phase. This is a much more complicated situation in heterogeneous domains because of capillary barrier effects. The relatively simple stochastic differential equations used in the single-phase case have to be modified to allow for NAPL pooling at heterogeneous interfaces. We do this by adding a jump term that is based on capillary diffusivity differences across the interface. Similarly, the upscaled dispersion term will have to be modified to reflect the effects of relative permeabilities and time dependent velocities. This is currently work in progress on which we will report. ~
DE: 1829 Groundwater hydrology
DE: 1832 Groundwater transport
DE: 1847 Modeling
DE: 1869 Stochastic hydrology
SC: Hydrology [H]
MN: Fall Meeting 2005