Elsevier

Powder Technology

Volume 413, January 2023, 118041
Powder Technology

Development of drag force model for predicting the flow behavior of porous media based on genetic programming

https://doi.org/10.1016/j.powtec.2022.118041Get rights and content

Highlights

  • A systematic laboratory investigation was carried out on different porous media by a self-made seepage apparatus.

  • A drag force model taking into account the main influencing factors was developed by genetic programing framework.

  • The effects of main factors on the flow characteristics were analyzed and discussed.

Abstract

Seepage in soils is a phenomenon related to the interaction between solid particles and fluid phase. The present study develops a drag force model by focusing on the voidage function using a genetic programing (GP) procedure. A systematic laboratory seepage tests was carried out on porous media with different materials by a self-made seepage apparatus. Based on the database obtained by the numerous seepage tests, the drag force model was developed with the aid of symbolic regression in genetic program. The results indicate that the developed drag force model by GP method composed of a constant and four gene items has satisfied performance in predicting the drag behavior of particles, which is attributed to the GP's advantages on optimizing both the parameters and structure of the model. Among the influencing factors, the gradation coefficient, porosity, and shape coefficient have a significant effect on the seepage characteristics of the porous media. The proposed model in this study could be used to analyze the flow characteristics of porous media in the field of geotechnical and ocean engineering.

Introduction

Seepage refers to the phenomenon that the interaction between fluid and solid particles when fluid flowing through the porous media. There are many engineering problems related to seepage. For example, the effective stress of soil will be reduced (in terms of the effective stress principle proposed by Terzaghi) and its structure could even be destroyed due to the effect of water flow on the particles [1]. The fine particles in the pores between larger particles could be taken away (i.e., piping phenomenon) when the flow velocity of water exceeds the critical value, which might bring high risk to the safety and stability of dam or foundation [2]. In addition, the evolution of river landforms and the ground deformation of artificial filling islands are closely related to erosion or scour [3]. Therefore, exploring the seepage problems, particularly the interaction between fluid and solid particles, is very vital for geotechnical, geological and ocean engineering.

There are many studies on the seepage of soil materials, and some related analytical solutions (theories or models) have been developed. When the flow is laminar and steady, there exists a linear relationship between the hydraulic gradient and flow velocity, and such correlation is the famous Darcy's law. For flow with a relatively larger velocity, the above relationship will gradually become nonlinear due to the increased flow resistance, which has been extensively investigated by many researchers [4,5]. For the nonlinear flow, Forchheimer [6] initially added a second-order quadratic term to the Darcy's law, which is widely accepted and used to predict the relationship between hydraulic gradient and flow velocity. Ergun [7] then modified the Forchheimer's equation by considering the effect of porous porosity and liquid viscosity. Subsequently, many researchers focused on the effect of particle size on the constant coefficients of Ergun's equation and made many modifications [[8], [9], [10], [11]].

The seepage of soil is actually the interaction between soil particles and fluid at the microscopic or mesocopic scale, and such local behavior of flow and the interaction between solid and liquid phases is involved in the hydrodynamic forces such as drag. The drag force means the combination of the normal (i.e., pressure) and tangential (i.e., wall shear stress) forces on the body in the flow direction. Many scholars have proposed series solutions to calculate the drag force and/or drag coefficient. For example, Khan and Richardson [12] compiled and analyze 300 data points from experimental results of various researchers, and then proposed a drag equation of a power form using nonlinear regression, which was improved by Brown and Lawler [13]. Afterwards, Khan and Richardson [14] suggested a method to calculate the drag force for sedimenting suspensions and fluidized beds of uniform spherical particles in liquids, and made a clarification on the relationship between the various drag coefficients for particles in concentrated suspensions. Barati et al. [15] provided a comprehensive literature review discussing the empirical or semi-empirical drag coefficient models of smooth spherical particles, and they classified them into two groups in terms of Reynolds number. Most of these drag force or drag coefficient models were established focusing on the single particle or fluidized beds composed of uniform spherical particles. However, many researchers have found that the drag force on a particle in a fluid-multiparticle interaction system is significantly different from that on an unhindered particle, when subjected to the same volumetric flux of fluid. Di Felice [16] introduced a voidage function to involve in the influence of surrounding particles on the drag force of single unhindered particle. Later on, many researches have proved that the Di Felice's method is not very accurate to predict the drag force of particle in the complicate system (the porous media with particles having different characteristics, e.g., porosity, grading and shape), and some scholars have made investigations on the reasons and solutions to reduce the errors [17]. With the aid of numerical Immersed Boundary Method, Tenneti et al. [18] and Tang et al. [19] proposed drag expressions for particles within assembles, respectively. On the other hand, most of the previous models for the estimation of the drag force or coefficient of particles only included individual or limited factors. Actually, the flow behavior of particles in the anfractuous porous media is very complex and can be influenced by many factors (e.g., particle shape, grading, porosity, etc.). Recently, Wang et al. [20,21] investigated the effect of particle shape on the drag force of highly irregular calcareous sands, and developed a drag force model considering such effect.

The above typical examples of the equations available in the literature for evaluating the drag force of particles are mainly based on essential assumptions and simplifications. In this case, these models thus have varying degree of accuracy in their application, depending very much on the number and quality of data used to derive them. Recently, the genetic programing method is being increasingly utilized in the geotechnical and hydraulic engineering. For instance, Hakimzadeh et al. [22] used genetic programing method to simulate the outflow hydrograph from earthen dam breach, and the results demonstrate that the results of the GP method are in good agreement with the observed values. Barati et al. [15] developed high accurate drag coefficient correlations from low to very high Reynolds numbers using a multi-gene genetic programming procedure. Mehr and Kahya [23] proposes a Pareto-optimal moving average multi-gene genetic programming (MA-MGGP) approach to develop a parsimonious model for single-station streamflow prediction. Genetic programing method could provide an effective solution in the multiscale and multiphase analyses. Unlike traditional regression analysis, genetic programming automatically evolves both the structure and parameters of the drag coefficient estimation model. Thus both parameters and structure of the model will be optimized [15].

The objective of this study is to develop an effective drag force model of particle in the porous media by focusing on the voidage function with the aid of genetic programming method. To this end, a series of seepage tests were carried out on porous media containing different particles by using a self-developed seepage apparatus, and total 106 sets of experimental data were obtained. Then the drag force model involved in five important influencing factors (i.e., equivalent particle size, porosity, shape coefficient, gradation coefficient, and Reynolds number) was developed based on the genetic programming framework. The effects of the main parameter on the flow behavior of porous media were analyzed with the aid of modified Ergun equation. Finally, the accuracy and universality of the developed GP model was validated using data from literatures as well as the experimental results in this study.

Section snippets

Mathematical equations

At present, there is rare universal theoretical model for the calculation of the particle drag force induced by fluid. Generally, the drag force is determined by the semi-empirical fitting equations. For spherical particles, the drag force, Fd0 can be calculated as follows,Fd0=18Cd0πρfufup2d2in which, Cd0 is the drag coefficient of a single spherical particle, ρf is the density of fluid (kg/m3), uf and up are the velocity of fluid and particle (m/s), respectively, d is the diameter of the

Comparison of hydraulic conductivity between experimental measurement and Ksingle bondC equation prediction

In order to verify the experimental results of the seepage tests, the Kozeny-Carman (Ksingle bondC) equation was used to make a comparison. The Ksingle bondC equation is a well-recognized equation to predict the saturated hydraulic conductivity of porous materials. It was first proposed by Kozeny [30] and later modified by Carman [31]. This equation was developed after considering a porous material as an assembly of capillary tubes for which the equation of Navier-Stokes can be used. The generalized Ksingle bondC equation can

The basic set-up of GP model

Based on the genetic programming method, the drag force model (i.e., voidage function) was inferred and obtained by taking into account the effects of multiple parameters, i.e., deq, ε, Ψ, C, Re in Eq. (9). In this study, the symbolic regression operation was used, and this is a method for searching the space of mathematical expressions, while minimizing various error metrics. Unlike traditional linear and nonlinear regression methods that fit parameters to an equation of a given form, symbolic

Verification of the GP model

To further verify the accuracy and universality of the developed GP model, 36 sets of data from literatures [38,39] were collected, and the basic properties of the test materials are summarized in Table 3. It should be noted that the information of environmental temperature of their tests was not given, so the effect of temperature was not considered. In addition, the flow direction of liquid during the seepage tests was horizontal, which was similar to that of the tests in this study. At the

Conclusions and remarks

In the present study, a systematic seepage test on non-cohesive granular materials with different properties was carried out. Based on the GP method, an effective drag force model focusing on the voidage function was developed by considering the main influencing factors of the porous media. According to the overall results of this study, the following conclusions can be drawn:

  • (1)

    The 5-parameter voidage function could efficiently reflect the drag force behavior of particles in the porous media. The

Latin

    A

    the input program data set

    Amp

    the area containing dl and dm

    Ap

    the surface area of the actual particle

    Asph

    the surface area of the volume equivalent sphere

    C

    the gradation coefficient

    Cc

    the curvature coefficient

    Cd0

    the drag coefficient of a single spherical particle

    CK-C

    the empirical coefficient of the Ksingle bondC equation

    Cu

    the nonuniform coefficient

    d

    the diameter of the spherical particle

    d50

    the particle size corresponding to the accumulated mass fraction in 50%

    deq

    the equivalent particle size of the irregular

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 51890912, 51879035) and the China Postdoctoral Science Foundation (2021M700672).

References (39)

  • P.U. Foscolo et al.

    A unified model for particulate expansion of fluidised beds and flow in fixed porous media

    Chem. Eng. Sci.

    (1983)
  • N. Epstein

    Comments on a unified model for particulate expansion of fluidized beds and flow in fixed porous media

    Chem. Eng. Sci.

    (1984)
  • Y. Wang et al.

    Artificial neural network model development for prediction of nonlinear flow in porous media

    Powder Technol.

    (2020)
  • T.J. Donohue et al.

    Improving permeability prediction for fibrous materials through a numerical investigation into pore size and pore connectivity

    Powder Technol.

    (2009)
  • Z. Li et al.

    Effects of particle diameter on flow characteristics in sand columns

    Int. J. Heat Mass Transf.

    (2017)
  • X. Zhang et al.

    Improving dam seepage prediction using back-propagation neural network and genetic algorithm

    Math. Probl. Eng.

    (2020)
  • J.H. van Lopik et al.

    The effect of grain size distribution on nonlinear flow behavior in sandy porous media

    Transp. Porous Media

    (2017)
  • P. Forchheimer

    Wasserbewegung durch boden

    Z. Ver. Deutsch. Ing.

    (1901)
  • S. Ergun

    Fluid flow through packed columns

    Chem. Eng. Prog.

    (1952)
  • Cited by (4)

    View full text