Non-equilibrium effect modeling and particle velocity estimation for particulate flow in porous media

The complex dynamics of fluid and particles flowing through pore space demands some relaxation time for particles to catch up with fluid velocity, which manifest themselves as non-equilibrium (NE) effects. Previous studies have shown that NE effects in particle transport can have significant consequences when relaxation time is comparable to the characteristic time associated with the fluid flow field. However, the existing models are lacking to account for this complicated relation between particles and fluid. In this paper, we adapt the general form of the harmonic oscillation equation to describe NE effects in the particulate flow system. The NE effect is evaluated by solving coupled mass balance equations with computational fluid dynamic (CFD) techniques using COMSOL Multiphysics((R)). A simplified straight-tube model, periodic converging-diverging tube model, and SEM image of a real pore network are applied in NE analyses. The results indicate that the time variation of the NE effect complies with the theory of stability. Two key parameters of the oscillator equation are amplitude (A) and damping ratio (zeta), where the former represents the magnitude of NE and the latter is an indication of flow path geometry. The velocity equations for particle transport in different flow path geometries are derived from the proposed NE equation, offering a quick estimation of particle velocity in the particulate flow. By conducting simulation on the SEM image of a real pore structure, the equivalent radii of the pores where particles move through were obtained. The outcome of this work can shed light on explaining the complex NE effects in porous media. The generalized equation to model NE can help temporarily decouple particle transport equations from fluid equations, facilitating much advanced particulate flow modeling problems in the large-scale problems.