Porous media: A model system for the physics of disorder

Transition from creeping via viscous-inertial to turbulent flow in fixed beds

This review is concerned with the analysis of flow regimes in porous media, in particular, in fixed beds of spherical particles used as reactors in engineering applications, or as separation units in liquid chromatography. A transition from creeping via viscous-inertial to turbulent flow is discussed based on macro-scale transport behaviour with respect to the pressure drop-flow rate dependence, in particular, the deviation from Darcy's law, as well as direct microscopic data which reflect concomitant changes in the pore-level hydrodynamics. In contrast to the flow behaviour in straight pipes, the transition from laminar to turbulent flow in fixed particulate beds is not sharp, but proceeds gradually through a viscous-inertial flow regime. The onset of this steady, nonlinear regime and increasing role of inertial forces is macroscopically manifested in the failure of Darcy's law to describe flow through fixed beds at higher Reynolds numbers. While the physical reasons for this failure still are not completely understood, it is not caused by turbulence which occurs at Reynolds numbers about two orders of magnitude above those for which a deviation from Darcy's law is observed. Microscopic analysis shows that this steady, nonlinear flow regime is characterized by the development of an inertial core in the pore-level profile, i.e., at increasing Reynolds number velocity profiles in individual pores become flatter towards the center of the pores, while the velocity gradient increases close to the solid-liquid interface. Further, regions with local backflow and stationary eddies are demonstrated for the laminar flow regime in fixed beds. The onset of local fluctuations (end of laminar regime) is observed at superficial Reynolds numbers on the order of 100. Complementary analysis of hydrodynamic dispersion suggests that this unsteady flow accelerates lateral equilibration between different velocities in fixed beds which, in turn, reduces spreading in the longitudial (macroscopic flow) direction.

Influence of stagnant zones on transient and asymptotic dispersion in macroscopically homogeneous porous media

The role of stagnant zones in hydrodynamic dispersion is studied for creeping flow through a fixed bed of spherical permeable particles, covering several orders of characteristic time and length scales associated with fluid transport. Numerical simulations employ a hierarchical model to cope with the different temporal and spatial scales, showing good agreement with our experimental results on diffusion-limited mass transfer, transient, and asymptotic longitudinal dispersion. These data demonstrate that intraparticle liquid holdup in macroscopically homogeneous porous media clearly dominates over contributions caused by the intrinsic flow field heterogeneity and boundary-layer mass transfer.

Numerical simulation and measurement of liquid hold-up in biporous media containing discrete stagnant zones

We have studied hydrodynamic dispersion in single-phase incompressible liquid flow through a fixed bed made of spherical, permeable (porous) particles. The observed behaviour was contrasted with the corresponding fluid dynamics in a random packing of impermeable (non-porous) spheres with an interparticle void fraction of 0.37. Experimental data were obtained in the laminar flow regime by pulsed field gradient nuclear magnetic resonance and were complemented by numerical simulations employing a hierarchical transport model with a discrete (lattice Boltzmann) interparticle flow field. Finite-size effects in the simulation associated with the spatial discretization of support particles or dimension and boundaries of the bed were minimized and the simulation results are in reasonable agreement with experiment.

Experimental evidence of power-law trapping-time distributions in porous media

We present experimental results of solute transport in porous samples made of packings of activated carbon porous grains. Exchange experiments, where the tagged solution initially saturating the medium is replaced with the same solution without tracer, are accurately described by macroscopic transport equations. On the other hand, in desorption experiments, where the tagged solution is replaced by water, the solute concentration exhibits a power-law decay for long times, which requires a more detailed, mesoscopic description. We reproduce this behavior within a continuous-time random-walk approach, where the waiting time distribution is related to the desorption isotherm. Results are compatible with a power-law trapping time distribution with divergent first moment, characteristic of anomalous (sub)diffusion.