The use of the inverse problem methodology in analysis of fluid flow through granular beds with non-uniform grain sizes

Wojciech Sobieski,Anna Trykozko

doi:10.31648/ts.6772

Abstract

The pressure drop during water flow through two gravel beds with 2-8 and 8-16 [mm] grain size was measured across a wide range of filtration velocities, and the optimal method for calculating the coefficients for Darcy’s law and Forchheimer’s law was selected. The laws and the experimental data were used to develop a computational program based on the Finite Element Method (FEM). The results were compared, and errors were analyzed to determine which law better describes flow data. Various methods of measuring porosity and average grain diameter, representative of the sample, were analyzed. The data were used to determine the limits of applicability of both laws. The study was motivated by the observation that computational formulas in the literature produce results that differ by several orders of magnitude, which significantly compromises their applicability. The present study is a continuation of our previous research into artificial granular materials with similarly sized particles. In our previous work, the results produced by analytical and numerical models were highly consistent with the experimental data. The aim of this study was to determine whether the inverse problem methodology can deliver equally reliable results in natural materials composed of large particles. The experimental data were presented in detail to facilitate the replication, reproduction and verification of all analyses and calculations.

Highlights

Fluid flow through porous media can be described mathematically at two qualitatively different levels
The main disadvantage of this approach is that constant values, which usually differ for various types of porous media, have to be incorporated into mathematical models
High percentage error does not confirm the above observation. These findings suggest that in forcedflow systems, such as the presented test stand, flow should be generally modeled with the Forchheimer equation or similar non-linear equations

Summary

Introduction

Fluid flow through porous media can be described mathematically at two qualitatively different levels. The first level (historical) involves macroscopic measurements. In this approach, a porous medium is regarded as a homogeneous medium where flow resistance is generally averaged in space and time (Ergun 1952, Hellström, Lundström 2006, Sidiropoulou et al 2007, Sobieski 2010). A generally applicable mathematical model for measuring flow resistance in porous media has never been developed. A dedicated methodology for calculating these coefficients has never been proposed, and dozens of formulas generating results that differ by many orders of magnitude have been described in the literature (Sobieski, Trykozko 2011)

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Conclusion