Markov processes for Advection-Diffusion Transport in Non-Smooth Media: Skew Brownian Motion and the Generalized Taylor-Aris Formula.
Abstract
Markov processes are used to study advective-diffusive transport of passive solutes in media with non-smooth diffusion tensor D and drift coefficient U. The classical Taylor-Aris formula for the effective dispersion in a cylinder is generalized to the case where U is bounded and D is bounded positive definite; both functions depending only on the transversal spatial coordinate. In the particular case of a two-dimensional arrangement of layers of constant D parallel to the direction of flow, we identify the Markov process modeling the motion of individual particles: in the transversal direction the motion is a generalization of skew Brownian motion; longitudinally the motion is an Itô process with paths depending on the transversal process. At an interface between layers where D takes two different values, the diffusing particles are more likely to go into the medium with the higher value of D. This feature is analytically characterized by the sample path properties of skew Brownian motion. The results are applied to give theoretical foundation to classical particle tracking methods for mass transfer in porous media.
- Publication:
-
AGU Fall Meeting Abstracts
- Pub Date:
- December 2007
- Bibcode:
- 2007AGUFM.H11C0655R
- Keywords:
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- 1829 Groundwater hydrology;
- 1832 Groundwater transport;
- 1849 Numerical approximations and analysis;
- 1869 Stochastic hydrology;
- 3265 Stochastic processes (3235;
- 4468;
- 4475;
- 7857)