Nonlinear Flow Behavior in Packed Beds of Natural and Variably Graded Granular Materials

Abstract

Under certain flow conditions, fluid flow through porous media starts to deviate from the linear relationship between flow rate and hydraulic gradient. At such flow conditions, Darcy's law for laminar flow can no longer be assumed and nonlinear relationships are required to predict flow in the Forchheimer regime. To date, most of the nonlinear flow behavior data is obtained from flow experiments on packed beds of uniformly graded granular materials (C u = d 60 /d 10 < 2) with various average grain sizes, ranging from sands to cobbles. However, natural deposits of sand and gravel in the subsurface could have a wide variety of grain size distributions. Therefore, in the present study we investigated the impact of variable grain size distributions on the extent of nonlinear flow behavior through 18 different packed beds of natural sand and gravel deposits, as well as composite filter sand and gravel mixtures within the investigated range of uniformity (2.0 < C u < 17.35) and porosity values (0.23 < n < 0.36). Increased flow resistance is observed for the sand and gravel with high C u values and low porosity values. The present study shows that for granular material with wider grain size distributions (C u > 2), the d 10 instead of the average grain size (d 50) as characteristic pore length should be used. Ergun constants A and B with values of 63.1 and 1.72, respectively, resulted in a reasonable prediction of the Forchheimer coefficients for the investigated granular materials.

Convective Acceleration in Porous Media

“…Particularly for the determination of the performance and the capacity of production wells and injection wells, i.e. in the petroleum industry (Huang and Ayoub 2008;Barree and Conway 2004), in groundwater wells utilized for domestic and irrigational purposes (Houben et al 2018;Wen et al 2011;van Lopik et al 2019), and for groundwater heat pump applications (Gjengedal et al 2021).…”

Quinn’s Law of Fluid Dynamics: Supplement #2 Reinventing the Ergun Equation

“…The value of the kinetic constant B=δ/2π (λ=1 for packed conduits) in Quinn's Law, whereas it has the constant value of 1.75 in the Ergun equation. Figure 8, we have plotted the experimental results for 10 micron silica particles reported in the Farkas paper after the Q-modified Ergun equation (10). Note that in this straight line plot the intercept represents the value of A and the slope the value of B.…”

“…In order to underscore the importance of our experimental protocol disclosed herein and to distinguish it from that used generally in the literature, we now focus on a worked example from a recent publication in the literature of a wellknown author of packed conduits, J. H. Van Lopik et al [9,10]. In both these papers, the authors are focused on the nonlinear flow behavior in packed conduits.…”

Influence of fluid viscosity and flow transition over non-linear filtration through porous media

“…Applicability of this equation depends on the modelling accuracy of the Darcy and non-Darcy coefBcients. Extensive experimental studies have been reported in the literature on the behaviour of these coefBcients with different parameters such as media size (Huang et al 2013;Salahi et al 2015;van Lopik et al 2017), degree of packing (Boomsma and Poulikakos 2002;Dan et al 2016;Houben et al 2018), grain size distribution (van Lopik et al 2020), convergent angle (Thiruvengadam and Kumar 1997;Venkataraman and Rao 2000;Rao 2004, 2006;Pasupuleti et al 2014;, pore geometry (Macini et al 2011;Shao et al 2020) Reynolds number (Sidiropoulou et al 2007), maximum pore diameter (Huang et al 2020), etc.…”