Stochastic Dynamics of Lagrangian Particles in Porous Media

Abstract

We present stochastic model for describing Lagrangian dynamics of a fluid particle in porous media. The model is based on the combination of the Langevin equation, which has viscous terms induced from Stokes' low and the effect of boundary layer around a fluid particle, and Bayes' theorem, and provides a probability density function (PDF) for velocity of fluid particles. The presented PDF describes fat tails of velocity distribution, which is often observed in fluid turbulence, and cut-off of the fat tails. We implemented a model experiment to obtain velocity distributions of tracer particles in porous media, which consists of glass beads and silicone gel. The velocity distributions are identified with the theoretical PDF and show different shapes depending on flow rate of silicone gel. Namely, stochastic dynamics of Lagrangian particles shows stochastic bifurcation in porous media. This result can be applied to investigate water pollution in soils and to develop inspection devices for water pollution with high accuracy.