Artificial neural network model development for prediction of nonlinear flow in porous media
Graphical abstract
Introduction
Porous media exist naturally, e.g., sand and rocks and can be constructed, e.g., packed sand particles, glass spheres, and other human-made particles. Study on fluid flow in porous media packed with stationary particles is of great importance to the applications of geotechnical, environmental and hydrological engineering, etc. [[1], [2], [3], [4]]. The study in this paper is motivated by the need for an appropriate model as an alternative approach to describe and predict the flow behaviour in porous media based on the hydraulic pressure and flow velocity.
When the flow is laminar and in steady state, a linear relationship between flow velocity and hydraulic gradient exists, and Darcy's law is applicable, as follows:where i is the hydraulic gradient (m/m), K is the hydraulic conductivity of the porous medium (m/s) and q is the specific discharge (m/s). This empirical linear relationship for fluid flow in porous media is only applicable for a specific regime of flow where flow velocities are sufficiently low, and it becomes less accurate as flow velocities become larger. The relationship between specific discharge and hydraulic gradient deviates from linear and gradually becomes nonlinear due to increased flow resistance at higher flow velocities, which has been extensively addressed in the literature [1,3,5]. For the nonlinear flow, a second order equation which is proposed by Forchheimer [5], is widely accepted and used to predict the relationship between hydraulic gradient and flow velocity. Forchheimer described the nonlinear behaviour of fluid flow by adding a second-order quadratic term to the Darcy's law:
Comparing Eqs. (1) and (2), a (s/m) is a parameter equal to the reciprocal of the hydraulic conductivity (i.e., a = 1/K) and b is the Forchheimer quadratic coefficient (s2/m2) depending on the characteristics of the porous media. Panfilov and Fourar [6] were the first to point out the main mechanism of the inertial term (bq2) relating to recirculation of eddies inside closed streamlines. This inertial term is empirical and is added to consider the microscopic inertial effects observed experimentally [1,7].
Numerous analytical solutions, numerical methods and computing programs have been established and available for the analysis of linear flow (Darcy flow). Similar tools are also available for the simulation of non-linear inertial flows (non-Darcy flow) [[7], [8], [9]], although the number is very limited. However, their application requires the understanding of the phenomenological coefficient a and b in Forchheimer equation (Eq. (2)). Extensive research have been conducted to obtain the Forchheimer equation by determining the expressions of the a and b. De Plessis and Masliyah [10] proposed a momentum transport equation by using volume averaging method for isotropic consolidated porous media. For the case without boundary effects, the equation reduced to a Forchheimer-type equation in which the permeability was a function of the porosity, tortuosity and a characteristic pore length. They concluded that the permeability in the Darcy flow regime was constant, which in the non-Darcy nonlinear flow, it decreased with increasing flow velocity. Schneebeli [11] proposed expressions of the coefficients a and b for porous media consisting of spheres. The average particle diameter is used as characteristic pore length of the porous media. Ward [12] proposed similar expressions based on nonlinear flow experiments with 20 different porous media consisting of granular materials ranging from glass beads to sand and gravel, having particle diameters of 0.27–16.1 mm. Sidiropoulou et al. [7] correlated the Forchheimer equation through determining the equation coefficient a and b based on an investigation of available experimental data in the literature. Based on the hydraulic radius theory of Kozeny-Carman for Darcy flow, Ergun [13] derived an expression for the b coefficient. Ergun's equation describes the relationship between pressure drop and flow velocity and it works very well for nonlinear flows in porous media of packed spherical particles. To a great extent, it is the most referred and used relationships [14,15] in the analyses of nonlinear flow. Many variations on the Ergun relations for different porous media with a wide range of Ergun constants are reported in the literature [1,3,16].
The above are typical examples of the equations available in the literature for evaluating the nonlinear flow in porous media by the Forchheimer equation. They are based on the assumptions and simplifications of the geometry of the pore spaces [[10], [11], [12]]. These equations thus have varying degree of accuracy in their application, depending also on the number and quality of data used to derive them [3,14]. For this it is necessary to develop new theoretical/empirical or semi-empirical predictive model to determine the Forchheimer coefficients. In traditional empirical or semi-empirical approaches, deterministic models concentrate on the physical meaning of the phenomenon only. A complete and successful deterministic model ought to represent all the phenomenological correlations that exist between the influencing factors (such as, particle properties) and flow behaviour (such as, pressure drop and flow velocity). In nonlinear flow analysis although the influence of the factors on the flow behaviours is well understood but there exists no definite correlations among those. So such model is not perceived to be the main choice in future study. Recently, the artificial neural network (ANN) method is being increasingly used in civil, environmental and hydraulic engineering [[17], [18], [19], [20], [21], [22]]. As for the specific science issue of fluid flow through porous media, the ANN also exhibits its advantages and applicability in multiscale and multiphase analyses, [[23], [24], [25], [26]]. In the ANN method, the relationship model between the dependent and independent variables are established by examining the patterns inherent within the data set. This is neither a mathematical model nor a physical model. Thus, the machine learning is conceived to be a good alternative and used in this study. However, application of the ANN method in analysis of the flow behaviour and estimation of the Forchheimer coefficients is very limited mainly due to lack of observed primary data for diverse types of porous media of particles.
The purpose of this study is to obtain determination of Forchheimer coefficients and to propose a predictive ANN model of nonlinear flow behaviour through a machine-learning process. To this end, a series of experimental investigations on the flow through packed column was performed in lab and a large set of pressure drop data from the experimental report in the literature was also collected. Within the data sets, various porous media consisting of particles ranging from glass beads to sand and crushed rock, having broad particle shapes and particle diameters have been covered. Based on these data sets, the important influencing factors on flow behaviour were identified and considered as the input parameters (i.e., particle density, particle diameter, particle shape and sample porosity) in the ANN structure. Different combinations of these input parameters were used to develop a robust network model structure for simulating the pressure drop-flow velocity relationship. The performance of the ANN model was evaluated by using multiple statistical criteria and it reveals the suitability of ANN approach for estimation of the Forchheimer coefficients and simulation of the nonlinear flow, which is always challenging for researchers. The proposed predictive approach by the machine learning process is considered as a good alternative way to analyse the nonlinear flow through porous media.
Section snippets
Pressure drop-flow velocity relationship
As mentioned above, Ergun's Eq. [13] which is the most referred and used one to described the relationship to between pressure drop and flow velocity, can be expressed byin which, Δp is the pressure drop (kPa) through the length (m), L, of the porous media in the flow direction; u is the average flow velocity (m/s); ε is the porosity of the porous media; ρf is the fluid density (kg/m3); μ is the fluid viscosity (Pa•s), deq is the equivalent particle diameter (m). X
Pressure drop-flow velocity relationship
Typical data for the modified pressure drop-flow velocity relationships from the flow experiments are shown in Fig. 5 according to Eq. (5) with different packed conditions (particle density, diameter, shape and sample porosity). To make a compatible comparison, the specific condition is varies with other conditions keeping constant. The general trend of the modified pressure drop is that it increases with flow velocity for the packed columns. As expected from a quadratic relationship when the
Conclusions
In the present study, based on a large set of experimental data for the flow through packed particle columns, the ANN prediction model of nonlinear flow behaviour was developed using the machine-learning algorithm. The performance of the established ANN model has been evaluated with multiple statistical criteria. Based on the overall results of this study, the following concluding remarks can be drawn:
- (1)
The importance of the influencing factors on the flow behaviour has been examined, and four
Notation
- The following symbols are used in this paper:
- i
hydraulic gradient;
- K
hydraulic conductivity of the porous medium, m/s;
- q
specific discharge, m/s;
- a
parameter equal to the reciprocal of the hydraulic conductivity, s/m;
- b
Forchheimer quadratic coefficient, s2/m2;
- Δp
pressure drop, kPa;
- L
length of the porous media in the flow direction, m;
- u
average flow velocity, m/s;
- ε
porosity of the porous media;
- ρf
fluid density, kg/m3;
- μ
fluid viscosity, Pa•s;
- deq
equivalent particle diameter, m;
- X
constants;
- Y
constants;
- M
constants;
- N
Credit author statement
Yin Wang: Conceptualization, Methodology, Writing-Review, Editing. Shixing Zhang: Data curation, Writing- Original draft preparation. Zhe Ma: Visualization, Investigation. Qing Yang: Writing-Reviewing, Supervision.
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
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China, No. 51890912; 51879035. The support from the National Key R&D Program of China (Project ID: 2016YFE0200100) is also acknowledged.
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