Development of drag force model for predicting the flow behavior of porous media based on genetic programming
Graphical abstract
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,in 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 KC equation prediction
In order to verify the experimental results of the seepage tests, the Kozeny-Carman (KC) equation was used to make a comparison. The KC 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 KC 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 KC 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).
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