Fluid flow through 3D-printed particle beds: a new technique for understanding, validating, and improving predictability of permeability from empirical equations

Abstract

The application of 3D technology for fabrication of artificial porous media samples improves porous media flow studies. The geometrical characteristics of a porous media pore channel: the channel shape, size, porosity, specific surface, expansion ratio, contraction ratio, and the tortuous pathway of the channel can be controlled through advanced additive manufacturing techniques (3D printing), computed tomography imagery (CT imaging) and image analysis methods. These 3D technologies have here been applied to construct and analyze four homogeneous porous media samples with predefined geometrical properties that are otherwise impossible to construct with conventional methods. Uncertainties regarding the geometrical properties are minimized because the 3D-printed porous media samples can be evaluated with CT imaging after fabrication. It is this combination of 3D technology that improves the data acquisition and data interpretation and contributes to new insight into the phenomenon of fluid flow through porous media. The effects of the individual geometrical properties on the fluid flow are then accounted for in permeability experiments in a Hassler flow cell. The results of the experimental work are used to test the theoretical foundation of the Kozeny–Carman equation and the extended version known as the Ergun equation. These equations are developed from analogies to the Hagen–Poiseuille flow equation. Based on the results from the laboratory experiments in this study, an analytical equation based on the analytical Navier–Stokes equations is presented as an alternative to the Hagen–Poiseuille analogy for porous media channels with non-uniform channel geometries. The agreement between experiment and the new equation reveals that the dissipating losses of mechanical energy in porous media flows are not a result of frictional shear alone. The mechanical losses are also a result of pressure dissipation that arise due to the non-uniformity of the channel geometry, which induced spatial variations to the strain rate field and induce acceleration of the velocity field in the flow through the porous medium. It is this acceleration that causes a divergence from linear flow conditions as the Stokes flow criterion (Re ≪ 1) is breached and causes the convective acceleration term to affect the flow behavior. The suggested modifications of theory and the presented experiments prove that the effects of surface roughness (1) do not alter the flow behavior in the Darcy flow regime or (2) in the Forchheimer flow regime. This implies that the flow is still laminar for the Forchheimer flow velocities tested.

Convective Acceleration in Porous Media

“…The permeability k and the inertial resistance factor β represent intrinsic geometrical properties that describe the size and shape of the porous media pore channels (Ruth and Ma 1992;Yazdchi and Luding 2012;Whitaker 1996;Gjengedal et al 2020). These geometrical properties are typically represented by a diameter of the solids that surround the pores (Chukwudozie and Tyagi 2013;van Lopik et al 2019).…”

“…However, the permeability and the inertial resistance factor should be identified with respect to the shape and size of the pores within the pore matrix. A new semi-analytical approach is presented in detail by Gjengedal et al (2020) and provides Eq. ( 2).…”

“…It is suggested that the C d depends on the constriction ratio and the streamlining shape of the pore channel, like the "minor loss" coefficient in the literature of fluid mechanics of pipes, Fig. 1 (Gjengedal et al 2020).…”

“…This paper explores how geometrical properties of a pore channel induced convective acceleration and how this affects the inertial losses. Recent laboratory studies on 3D fabricated particle beds with predefined pore geometries enables more detailed and controlled analysis of the geometrical properties that are important for porous media flows (Huang, et al 2013;Gjengedal et al 2020). Gjengedal et al (2020) has hypothesized that porous media flows exhibit similar behavioural traits to that of pipe flows in situations where the pipe channels are nonuniform, the so called minor losses approach in fluid mechanics.…”

“…Recent laboratory studies on 3D fabricated particle beds with predefined pore geometries enables more detailed and controlled analysis of the geometrical properties that are important for porous media flows (Huang, et al 2013;Gjengedal et al 2020). Gjengedal et al (2020) has hypothesized that porous media flows exhibit similar behavioural traits to that of pipe flows in situations where the pipe channels are nonuniform, the so called minor losses approach in fluid mechanics. A method of comparison with the pipe flow equations is here presented, and the similarities between the porous media flows and pipe flows are further assessed.…”